Canonical labelling of random regular graphs

Canonical lab elling of random regular graphs Mikhail Isaev ∗ T am´ as Mak ai † Brendan McKa y ‡ P a w e l Pra lat § Jane T an ¶ Maksim Zh uk o vskii ‖ Abstract W e pro ve that whenever d = d ( n ) → ∞ and n − d → ∞ as n → ∞ , then with high probability for any non-trivial initial colouring, the colour reﬁnement algorithm distinguishes all v ertices of the random regular graph G n,d . This, in particular, implies that with high probabilit y G n,d admits a canonical lab elling computable in time O (min { n ω , nd 2 + nd log n } ), where ω < 2 . 372 is the matrix m ultiplication exponent. 1 In tro duction Giv en an input graph G , a c anonic al lab el ling algorithm computes a bijection π G : V ( G ) 7→ { 1 , . . . , n } with the following prop ert y: if a graph G ′ is isomorphic to G , then the relab elled v ersions of G and G ′ , under the actions of π G and π G ′ , are identical. There is a linear time reduc- tion from the graph isomorphism problem to canonical lab elling: once the lab ellings π G , π G ′ ha v e b een computed for tw o input graphs G, G ′ , it takes time O ( | E ( G ) | ) to chec k whether G, G ′ are isomorphic. The b est known (in the worst case) algorithm for the graph isomorphism problem is due to Babai [ 7 , 28 ]: it runs in time exp( O (log 3 n )) for n -v ertex graphs. A quasi-polynomial bound exp(log O (1) n ) is also known for the canonical lab elling problem [ 8 ]. Nevertheless, for graphs with b ounded degree, b oth problems can be solv ed in p olynomial time [ 10 , 40 ]. In particular, this is the case for d -regular graphs when d = const . In this pap er, w e sho w that there exists a p olynomial-time canonical lab elling algorithm for almost al l d -regular graphs for al l 0 ≤ d ≤ n − 1. Colour reﬁnemen t (CR) is a simple algorithmic routine that operates on v ertex-coloured graphs. F or an input graph G with initial colouring C 0 : V ( G ) → Z , CR iterativ ely computes new colourings. A t round t , C t ( v ) is a pair ( C t − 1 ( v ) , C t − 1 ( N ( v ))), where C t − 1 ( N ( v )) is the m ultiset of C t − 1 -colours of neigh b ours of v . That is, the process reﬁnes the initial partition C 0 and halts once the partition stabilises. Let us call a colouring discr ete if every pair of v ertices is coloured diﬀerently . If CR runs on an uncoloured graph (i.e., there is only one initial colour) and outputs a discrete colouring, then, since the v ertex colours are isomorphism-in v arian t, this yields a canonical labelling by n umbering the colour names in the lexicographic order. In [ 9 , 19 , 26 ], it w as pro ved that CR run on a binomial random graph G ( n, p ) 1 outputs a discrete colouring with high probabilit y ( whp , in what follows) 2 ∗ Sc ho ol of Mathematics and Statistics, UNSW Sydney , Sydney , NSW, 2052, Australia † Mathematisc hes Institut der Universit¨ at M ¨ unc hen, Munich D-80333, German y ‡ Sc ho ol of Computing, Australian National Universit y , Canberra, ACT, 2601, Australia § Departmen t of Mathematics, T oron to Metrop olitan Universit y , T oronto, ON M5B 2K3, Canada ¶ Mathematical Institute, Universit y of Oxford, Oxford OX2 6GG, UK ‖ Sc ho ol of Computer Science, The Universit y of Sheﬃeld, Sheﬃeld S1 4DP , UK 1 The vertex set of G ( n, p ) is { 1 , . . . , n } , and each pair of vertices is adjacent with probabilit y p = p ( n ), indep en- den tly of the other pairs. 2 A sequence of even ts B n holds with high probability if P ( B n ) → 1 as n → ∞ . 1 whenev er (1 + ε ) ln n n < p ≤ 1 2 , which implies a near linear time algorithm 3 for canonical lab elling of G ( n, p ) [ 13 ]. W e stress that for regular uncoloured graphs G , CR terminates immediately with a trivial colouring, and is therefore unsuitable for canonical lab elling. F or a p ositiv e integer n , we denote [ n ] := { 1 , . . . , n } . Let d ≤ n − 1 b e a non-negative integer such that dn is even. Let G n,d b e a uniform distribution o ver all d -regular graphs on [ n ]. W e write G n ∼ G n,d for a graph sampled from this distribution, i.e. G n is a uniformly random d -regular graph on [ n ]. Since, for constant d , eﬃcien t canonical lab elling algorithms are known, we fo cus on the case d = ω (1). Moreo v er, since the edge complement of a d -regular graph is ( n − 1 − d )-regular, the edge complemen t of G n,n − 1 − d is distributed as G n,d . Therefore, we ma y restrict ourselv es to d ≤ n/ 2. Our main result shows that the trivialit y of the initial colouring is the only obstacle for a complete reﬁnement: once a non-trivial initial colouring is pro duced, whp CR run on G n ∼ G n,d outputs a discrete colouring whic h is suitable for canonical labelling. Before we state the main result of this pap er, we need one more deﬁnition. F or a connected graph G , w e denote by diam( G ) its diameter. Theorem 1. L et d 0 b e lar ge enough, let d = d ( n ) b e such that d 0 ≤ d ≤ n/ 2 , and let G n ∼ G n,d . Then, the fol lowing holds whp: for every non-trivial p artition [ n ] = V 1 ⊔ V 2 of the vertex set of G n , CR runs at most 2 diam( G n ) + 3 steps on G n and outputs a discr ete c olouring. Remark 2. W e did not try to optimise the b ound on the num b er of rounds, and we b eliev e that 2 diam( G n ) + 3 is sub optimal. In particular, for d ≥ n 1 / 2+ ε , we show that 2 diam( G n ) + 1 = 5 rounds is enough. Actually , it is natural to susp ect that the total n um b er of rounds needed is (1 + o diam( G n ) (1)) diam( G n ) whp. W e also note that our technique do es not directly generalise to small v alues of d ; in particular, there app ears to b e a natural barrier at d = 3. Nevertheless, we b eliev e that the statement of Theorem 1 can b e extended to all 3 ≤ d ≤ n/ 2. W e now describ e a p ossible approac h to canonical lab elling of G n ∼ G n,d based on our main result. Since the output of CR can b e computed in time O (( n + | E ( G ) | ) log n ) on an n -vertex graph G [ 13 ], Theorem 1 reduces the problem to eﬃciently ﬁnding an isomorphism-preserving partition of [ n ]. T o this end, recall that, for an y d = ω (1), whp G n con tains a triangle [ 32 ]. Let t i b e the n um b er of triangles that con tain the vertex i ∈ [ n ] in G n , and let t = max { t 1 , . . . , t n } . Using fast square matrix multiplication, it is p ossible to compute the vector ( t 1 , . . . , t n ) in time n ω , where ω < 2 . 372 [ 1 ]. Alternativ ely , this vector can b e computed in time O ( nd 2 ) using the standard triangle-listing algorithm [ 17 ], which is faster when d < n 0 . 685 , assuming the b est known upp er b ound on the matrix m ultiplication exp onen t ω (and yields the b ound o ( n 2 ) for d = o ( √ n ), which is faster than any algorithm based on square matrix multiplication since ω ≥ 2). Then, we can partition [ n ] into sets V 1 , V 2 where V 1 con tains all vertices i for which t i = t and V 2 con tains the rest. It remains to show that this partition is non-trivial. Since whp G n con tains a triangle, V 1 is non-empty . When d = o ( n 1 / 3 ), the num b er of triangles is sublinear whp (b y Theorem 11 b elow), which immediately implies that there are vertices that do not b elong to a triangle, and so the set V 2 is non-empty as well. In order to show that V 2 is non-empty (i.e., that there are t w o vertices u, v with t u  = t v ) whp when d = ω ( n 1 / 3 ), ﬁx tw o v ertices u, v and exp ose their neigh b ourho o ds N ( u ) , N ( v ). Then exp ose the edges inside N ( u ). Assuming t u = t v , the set of exp osed edges identiﬁes the num b er of edges in E ( G n [ V ]) \ E ( G n [ U ]). By standard coun ting argumen ts (or using switc hings), it follows directly that, for any ﬁxed x = x ( n ) ∈ Z ≥ 0 , the probabilit y that the latter set con tains exactly x edges tends to zero. Finally , when d = Θ( n 1 / 3 ), 3 There exists an implementation of CR that runs in time O (( n + | E ( G ) | ) log n ) on any n -vertex graph G [ 13 ]. Ho wev er, whp CR halts on G ( n, p ) in O (1) rounds, implying a linear time b ound O ( n 2 p ). 2 it is kno wn that the num b er of triangles X n in G n satisﬁes a cen tral limit theorem: X n − E X n √ V ar X n con v erges in distribution to a standard normal random v ariable as n → ∞ [ 22 ]. In particular, if P ( ∀ i t i = t ) > ε , then P ( X n /n ∈ Z ) > ε , contradicting the central limit theorem. Therefore, we get the following. Corollary 3. L et d = d ( n ) b e such that d → ∞ and n − d → ∞ , and let G n ∼ G n,d . Ther e exists an algorithm that runs in time O (min { n ω , nd 2 + nd log n } ) on d -r e gular n -vertex gr aphs and whp outputs a c anonic al lab el ling of G n . Related work. In [ 14 , 37 ], it is prov ed that, when d ≤ n ε , for a suﬃciently small ε , then with high probabilit y G n ∼ G n,d admits a canonical lab elling via the 2-dimensional W eisfeiler-Leman algorithm (2-WL) [ 50 ], whic h is a generalisation of CR where the colouring is applied to pairs of v ertices, and whose running time is O ( n 3 log n ) [ 29 ]. The weak er v ersion of 2-WL suggested b y Boll´ obas [ 14 ] canonically lab els G n in time O ( n 3 / 2+ ε ) whp. Moreov er, Ku ˇ cera [ 37 ] claimed that his v ersion of the algorithm runs in av erage time O ( nd ) for d = O (1) which we failed to v erify 4 . W e therefore conclude this paragraph by asking whether there exists a linear-time algorithm — running in time O ( nd ) on d -regular n -vertex graphs — that canonically lab els G n whp, at least for some 3 ≤ d ≤ n/ 2 (either for all d = O (1), where the hidden constant factor does not dep end on the time complexit y , or for some d = ω (1)). F or binomial random graphs G ( n, p ), the study of canonical lab elling algorithms has b een more extensiv e. Babai, Erd˝ os, and Selko w [ 9 ] prov ed that CR outputs a canonical labelling of G ( n, 1 / 2) in linear time whp since it p erforms only a b ounded num b er of reﬁnemen t steps. The argumen t of [ 9 ] can b e extended to show [ 15 , Theorem 3.17] that the CR colouring of G ( n, p ) is whp discrete for all n − 1 / 5 ln n ≪ p ≤ 1 / 2. Bollob´ as [ 14 ] show ed a p olynomial time canonical lab elling algorithm for (1 + ε ) ln n n ≤ p ≤ 2 n − 11 / 12 , which is a weak er version of 2-WL. The next improv ement was obtained b y Cza jk a and P andurangan [ 19 ]: they extended the range of applicabilit y of CR to ln 4 n n ln ln n ≪ p ≤ 1 2 , whic h was ﬁnally extended to (1 + ε ) ln n n ≤ p ≤ 1 2 b y Gaudio, R´ acz, and Sridhar [ 26 ]. Linial and Mosheiﬀ [ 39 ] sho w ed that 2-WL outputs canonical lab elling of G ( n, p ) whp when 1 n ≪ p ≤ 1 2 . Finally , a p olynomial time algorithm that lab els canonically G ( n, p ) with high probability for all 0 ≤ p = p ( n ) ≤ 1 w as independently established in [ 6 , 48 ], with CR as the main ingredien t (note that b elo w the connectivity threshold ln n n , whp G ( n, p ) con tains many isolated vertices, so CR-colouring is not discrete). A partition [ n ] = V 1 ⊔ . . . ⊔ V t of the vertex set of a graph G is called e quitable if, for any 1 ≤ i, j ≤ t , not necessarily diﬀeren t, an y t w o vertices in V i ha v e exactly the same num b er of neigh b ours in V j . Clearly , if a graph G admits a non-trivial equitable partition V 1 ⊔ . . . ⊔ V t , then CR do es not reﬁne it. The opp osite statemen t is also true — if there is a partition that CR do es not reﬁne, then this partition is equitable. Theorem 1 implies that whp CR reﬁnes an y non-trivial partition of G n,d with any n um b er of parts. Therefore, it implies that whp G n ∼ G n,d do es not ha v e an equitable partition other than those with 1 part or n parts. If a graph G has a non- trivial automorphism group Aut( G ), then for any non-trivial automorphism σ ∈ Aut( G ), its cycle 4 The pap er do es not pro vide a pro of of this fact. How ever, the algorithm computes the vertices on shortest cycles as a subroutine and uses the following assertion [ 37 , Theorem 3.1]: All cycles of the length k in d -regular graph can b e found in time O ( n min { n, ( d − 1) k/ 2 } ). First, we believe that the factor ( d − 1) k/ 2 should instead read ( d − 1) ⌈ k/ 2 ⌉ (for instance, it is unclear how all triangles could b e found in time nd 1 . 5 — say , in a union of ( d + 1)-cliques, there are Θ( nd 2 ) triangles, which gives the low er bound Ω( nd 2 ) on time needed to list them). Second, this b ound (even with the fractional p ow er) is not enough to get the exp ected time O ( nd ). Indeed, G n has a triangle with asymptotic probability 1 − exp( − ( d − 1) 3 / 6). So w e get the b ound on the expected time to b e at least (1 − exp( − ( d − 1) 3 / 6) − o n (1)) nd 1 . 5 ∼ nd 1 . 5 as d → ∞ . 3 decomp osition iden tiﬁes an equitable partition of G . Therefore, if a graph do es not ha ve a non- trivial equitable partition, then its group of automorphisms is either trivial or cyclic. Ho w ev er, in the second case the graph must b e vertex-transitiv e, whic h is not the case for G n whp. In particular, as noted ab o v e, whp there exist tw o vertices that b elong to diﬀeren t num b ers of triangles. As a result, Theorem 1 implies that G n is asymmetric whp for all d such that d → ∞ and n − d → ∞ . This is a kno wn result for the en tire range 3 ≤ n ≤ d − 4, ha ving b een ﬁrst established for d = o ( √ n ) in [ 44 ] and then for d ≫ log n in [ 31 ]. Pro of strategy . The key step to obtain Theorem 1 is to show that after some num b er of rounds of CR, one can coarsen the partition asso ciated with colours to get a partition with k parts of comparable size (for an y desired arbitrarily large constant k ). The following statement makes this precise. Note that w e may coarsen a partition at an y stage of the CR algorithm if it is conv enient for the argument that follows. Clearly , one may couple the original pro cess with the mo diﬁed one so that the partitions in the original pro cess are reﬁnement of the corresp onding partitions in the mo diﬁed one. In particular, if the mo diﬁed pro cess reaches discrete colouring, then so do es the original one. Theorem 4. L et d 0 b e lar ge enough, let d 0 ≤ d = d ( n ) ≤ n/ 2 , and let G n ∼ G n,d . L et k ∈ N b e an arbitr ary c onstant. Then, the fol lowing holds whp: for every non-trivial p artition [ n ] = V 1 ⊔ V 2 of the vertex set of G n , after diam( G n ) + 2 r ounds of CR, ther e exists a p artition [ n ] = V 1 ⊔ . . . ⊔ V k such that for any i ∈ [ k ] , n/ 3 k ≤ | V i | ≤ 3 n/k and V i is a union of some c olour classes (in other wor ds, V 1 ⊔ . . . ⊔ V k is a c o arsening of the p artition asso ciate d with r esulting c olour classes). Once there are many parts of linear and comparable sizes, CR distinguishes all vertices after some additional n umber of rounds. Theorem 5. L et d 0 b e lar ge enough, let d 0 ≤ d = d ( n ) ≤ n/ 2 , and let G n ∼ G n,d . Ther e exists some universal lar ge c onstant k ∈ N such that the fol lowing holds whp: for every initial p artition [ n ] = V 1 ⊔ . . . ⊔ V k such that for any i ∈ [ k ] , n/ 3 k ≤ | V i | ≤ 3 n/k , CR terminates on G n after at most diam( G n ) + 1 r ounds and outputs a discr ete c olouring. Theorem 1 follows immediately from Theorem 4 and Theorem 5 . T o prov e each of these t w o theorems, w e address t wo regimes for d separately: the dense case with d ≥ n 1 / 2+ ε , and the sparse case with d 0 ≤ d ≤ n 10 / 17 . The corresp onding statements are reiterated in Sections 3 , 4 , 5 and 6 . Theorem 4 for d ≥ n 1 / 2+ ε is prov ed in Section 3 . Its pro of consists of tw o parts. First, we show that whp after one reﬁnement round there exists a coarsening [ n ] = U 1 ⊔ U 2 of the CR-partition suc h that | U 1 | , | U 2 | ≫ n/d ( Theorem 18 ). One more round is needed to get all colour classes of size at most δ n , for an arbitrary constan t δ > 0 ( Theorem 19 ). The latter claim follows from the fact that there is no large set with all vertices having the same degree proﬁle with resp ect to ( U 1 , U 2 ). This is the main tec hnical complication in the pro of of Section 3 in the dense case: although this fact is easy to show in G ( n, p ), in random regular graphs w e cannot rely on lo cal limit theorems. Instead we use asymptotic estimations of the num b er of graphs with a given degree sequence as w ell as an ti-concen tration prop erties of the h yp ergeometric distribution. The sparse case d = o ( n ) is addressed in Section 4 . Here, we show that, for every initial colouring V 1 ⊔ V 2 , where | V 1 | < cn , for a suﬃcien tly small constan t c > 0, after a few rounds of colour reﬁnemen t, we will get a union of colour classes U of size | U | ∈ [ cn/d, cn ] ( Theorem 16 ). One more round is needed to get a set U ′ of size | U ′ | ∈ [ nℓ/d, 0 . 999 n ], for an arbitrarily large constan t ℓ ( Theorem 25 ). Then, similarly to the dense case, we sho w that there is no set of size more than δ n such that all its v ertices hav e same n um b er of neighbours in U ′ , that giv es us the desired partition ( Theorem 23 ). All three lemmas 4 rely on switching arguments. In particular, the last tw o lemmas use switc hings to establish an analogue of the Erd˝ os–Littlewoo d–Oﬀord theorem in the context of uniformly random graphs with a ﬁxed degree sequence. Theorem 5 is pro ved in Sections 5 and 6 . The dense case is signiﬁcan tly easier. F or instance, when d = Θ( n ), t w o reﬁnement rounds are enough to obtain a discrete colouring whp. Indeed, let u, v b e tw o ﬁxed vertices. Exp ose the neighbourho o ds N ( u ) , N ( v ), and all the edges that touch N ( v ). The exp osed edges identify degree proﬁles of vertices in N ( u ) \ N ( v ) with resp ect to the ﬁxed partition. Since the latter set has size Θ( n ), it is extremely unlikely that all the degrees are equal to the ﬁxed v alues. Clearly , the probability of this even t is  1 / p n/k  Θ( kn ) in G ( n, p ), which is enough to ov ercome the union b ound with ro om to spare. In order to transfer this b ound to random regular graphs, we use asymptotic en umeration of graphs with a given degree sequence and an ti-concen tration inequalities for hypergeometric distribution, as for the dense case in Theorem 4 . F or n 1 / 2+ ε ≤ d = o ( n ), w e need one additional reﬁnement round in order to reac h a set of v ertices at distance at most 2 from { u, v } of size Θ( n ). The sparse case d ≤ n 10 / 17 , addressed in Section 6 , requires a more delicate switc hing argument and constitutes the most technical part of the pap er. Here, in order to reach a set of size Θ( n ), from ﬁxed vertices u, v , we need diam( G n ) rounds. Then, in con trast to the sparse case in Theorem 4 , w e need a multidimensional analogue of the Erd˝ os– Littlew o od–Oﬀord theorem ( Theorem 32 ), since the degree proﬁles are considered with resp ect to k sets of the partition. Nev ertheless, the claim can still be established by applying a similar switc hing argumen t Θ( k ) times. Organisation. W e start by presenting some preliminary results on properties of random graphs and concentration inequalities in Section 2 . The rest of the pap er is devoted to the pro of of Theorems 4 and 5 which immediately imply our main Theorem 1 . Theorem 4 is prov ed across Sections 3 and 4 where the dense and sparse c ases are treated resp ectively . Sections 5 and 6 are dev oted to the dense and sparse cases of Theorem 5 . Finally , in Section 7 we present a pro of of the expansion Theorem 16 that asserts that whp for ev ery set, its size remains concen trated after multiple expansion rounds. This lemma ma y b e of indep enden t interest and useful in other con texts. Notation. F or a graph G , a set of vertices U ⊆ V ( G ), and a non-negative in teger r , w e denote b y S r ( U ) the sphere of radius r around U in the graph metric, omitting the dep endency on G since the underlying graph is alwa ys clear from the context. That is, S r ( U ) consists of v ertices v suc h that the length of a shortest path from v to U equals r . In particular, S 0 ( U ) = U . W e also denote B r ( U ) = ∪ 0 ≤ i ≤ r S i ( r ) the ball of radius r around U . W e sometimes denote S 1 ( U ) by N ( U ) and refer to it as the neighbourho o d of U . F or a set of vertices X and a v ertex x / ∈ X , w e denote b y N X ( x ) the num b er of neighbours of x in X . W e also use the standard notation G [ U ] for the subgraph of G induced by a set U ⊆ V ( G ), and G [ U × V ] for the bipartite subgraph with (disjoin t) parts U and V , consisting of all edges of G with one endpoint in U and the other in V . F or a giv en degree sequence d = ( d 1 , . . . , d n ), we will use g ( d ) to denote the n umber of graphs on the v ertex set [ n ] with the degree sequence d . W e often write A ⊔ B to denote the union of t w o disjoint sets A and B . Finally , for a random v ariable X with distribution Q , we write X ∼ Q . In particular, X ∼ Bin( n, p ) is a binomial random v ariable with n trials and success probability p . 5 2 Preliminaries In this section, we collect some probabilistic to ols as well as prop erties of random regular graphs that we will use in the main pro ofs to follow. 2.1 Concen tration Inequalities W e will use the follo wing sp eciﬁc instances of Chernoﬀ ’s b ound — see, for example, [ 30 , Theo- rem 2.1]. Let X ∼ Bin( n, p ). Then, for any t ≥ 0 we hav e P ( X ≥ E X + t ) ≤ exp  − t 2 2( E X + t/ 3)  (1) P ( X ≤ E X − t ) ≤ exp  − t 2 2 E X  . (2) Remark 6. The same b ounds hold for a random v ariable with the hypergeometric distribution with parameters N , n , and m (see, for example, [ 30 , Theorem 2.10]). 2.2 Prop erties of Binomials No w, let us start with a few auxiliary observ ations. Lemma 7. F or al l p ositive inte gers b 1 < a 1 , b 2 < a 2 , a a b b ( a − b ) a − b ≥ a a 1 1 a a 2 2 b b 1 1 ( a 1 − b 1 ) a 1 − b 1 b b 2 2 ( a 2 − b 2 ) a 2 − b 2 , wher e a = a 1 + a 2 and b = b 1 + b 2 . Pr o of. Let us ﬁx non-negative integers b 1 < a 1 , b 2 < a 2 . Let a = a 1 + a 2 and b = b 1 + b 2 . W e start with the follo wing inequality , which follows immediately from the V andermonde’s identit y: for any n ∈ N ,  na nb  1 /n = nb X k =0  na 1 k  na 2 nb − k  ! 1 /n ≥  na 1 nb 1  1 /n  na 2 nb 2  1 /n . (3) Using Stirling’s formula ( s ! = (1 + o (1)) √ 2 π s ( s/e ) s ), the left hand side of ( 3 ) can b e estimated as follo ws:  na nb  1 /n =  ( na )! ( nb )! · ( n ( a − b ))!  1 /n =  (1 + o (1)) √ 2 π na ( na/e ) na √ 2 π nb ( nb/e ) nb · p 2 π n ( a − b )( n ( a − b ) /e ) n ( a − b )  1 /n = a a b b ( a − b ) a − b  (1 + o (1)) a 2 π nb ( a − b )  1 / 2 n = a a b b ( a − b ) a − b exp  − Θ  log n n  → a a b b ( a − b ) a − b , 6 as n → ∞ . The right hand side of ( 3 ) can b e dealt with the same wa y:  na 1 nb 1  1 /n  na 2 nb 2  1 /n → a a 1 1 b b 1 1 ( a 1 − b 1 ) a 1 − b 1 · a a 2 2 b b 2 2 ( a 2 − b 2 ) a 2 − b 2 , as n → ∞ . This veriﬁes the desired inequalit y . The preceding lemma has a useful corollary , which w e record as follows. Corollary 8 (Anti-concen tration of hypergeometric distribution) . F or inte gers 0 < b 1 < a 1 , 0 < b 2 < a 2 ,  a 1 b 1  a 2 b 2  ≤  a b  , (4) and  a 1 b 1  a 2 b 2  ≤ 2 3 s b ( a − b ) a 1 a 2 ab 1 ( a 1 − b 1 ) b 2 ( a 2 − b 2 )  a b  , (5) wher e a = a 1 + a 2 and b = b 1 + b 2 . In p articular, for al l inte gers 0 < b < a and p ositive inte gers k ,  a b  k ≤  a b ( a − b )  ( k − 1) / 2  k a k b  . (6) Mor e over, for al l inte gers 0 < b < a ,  2 a 2 b  ≤ 4 r b ( a − b ) a  a b  2 ≤ 2 √ a  a b  2 . (7) Pr o of. Let us ﬁx p ositiv e in tegers b 1 < a 1 , b 2 < a 2 . Let a = a 1 + a 2 and b = b 1 + b 2 . The ﬁrst inequalit y is a trivial consequence of V andermonde’s inequality . W e will use the following v ariant of the Stirling’s form ula that holds for all positive in tegers s : √ 2 π s  s e  s < √ 2 π s  s e  s exp  1 12 s + 1  < s ! < √ 2 π s  s e  s exp  1 12 s  ≤ e 1 / 12 √ 2 π s  s e  s . Using these inequalities and Theorem 7 , w e get  a 1 b 1  a 2 b 2  ≤ e 1 / 12 √ 2 π a 1 ( a 1 /e ) a 1 √ 2 π b 1 ( b 1 /e ) b 1 p 2 π ( a 1 − b 1 )(( a 1 − b 1 ) /e ) a 1 − b 1 · e 1 / 12 √ 2 π a 2 ( a 2 /e ) a 2 √ 2 π b 2 ( b 2 /e ) b 2 p 2 π ( a 2 − b 2 )(( a 2 − b 2 ) /e ) a 2 − b 2 = e 1 / 6 2 π · r a 1 a 2 b 1 ( a 1 − b 1 ) b 2 ( a 2 − b 2 ) · a a 1 1 a a 2 2 b b 1 1 ( a 1 − b 1 ) a 1 − b 1 b b 2 2 ( a 2 − b 2 ) a 2 − b 2 ≤ e 1 / 6 2 π · r a 1 a 2 b 1 ( a 1 − b 1 ) b 2 ( a 2 − b 2 ) · a a b b ( a − b ) a − b = e 1 / 3 √ 2 π · s b ( a − b ) a 1 a 2 ab 1 ( a 1 − b 1 ) b 2 ( a 2 − b 2 ) · √ 2 π a ( a/e ) a e 1 / 6 √ 2 π b ( b/e ) b p 2 π ( a − b )(( a − b ) /e ) a − b ≤ 2 3 s b ( a − b ) a 1 a 2 ab 1 ( a 1 − b 1 ) b 2 ( a 2 − b 2 )  a b  , 7 since e 1 / 3 / √ 2 π ≈ 0 . 557 < 2 / 3, proving ( 5 ). T o see that ( 6 ) follows, note that for an y non-negative in tegers a < b and positive in teger i ,  a b  ia ib  ≤ 2 3  ( i + 1) a bi ( a − b )  1 / 2  ( i + 1) a ( i + 1) b  ≤  a b ( a − b )  1 / 2  ( i + 1) a ( i + 1) b  , since 2 3 p ( i + 1) /i ≤ 2 √ 2 / 3 ≈ 0 . 943 < 1. Applying the ab o v e inequalit y k − 1 times, we get the desired conclusion. Finally , for ( 7 ), note that  2 a 2 b  ≤ e 1 / 12 p 2 π (2 a )(2 a/e ) 2 a p 2 π (2 b )(2 b/e ) 2 b p 2 π (2( a − b ))(2( a − b ) /e ) 2( a − b ) = e 5 / 12 √ π · r b ( a − b ) a · √ 2 π a ( a/e ) a e 1 / 6 √ 2 π b ( b/e ) b p 2 π ( a − b )(( a − b ) /e ) a − b ! 2 ≤ 4 r b ( a − b ) a ·  a b  2 ≤ 2 √ a ·  a b  2 , since e 5 / 12 √ π ≈ 2 . 689 < 4, which completes the pro of of the corollary . 2.3 Coun ting Graphs Recall that, we denote the num b er of graphs on the vertex set [ n ] with a given degree sequence d b y g ( d ). The following result gives tight asymptotic b ounds on g ( d ) for any degree sequence satisfying some mild condition. Dense graphs were inv estigated in [ 42 ] but the result was generalised to sparser graphs in [ 38 ]. The result b elo w states the bound in a slightly diﬀeren t form, whic h is more conv enient for our purp oses. W e provide a short proof of this reformulation. Theorem 9 ([ 38 , 42 , 43 ], rephrased) . L et d = ( d 1 , . . . , d n ) b e any de gr e e se quenc e such that P n i =1 d i is even and for al l i ∈ [ n ] , | d i − d | ≤ d 1 / 2+ ε ′ for some ε ′ > 0 , wher e d is the aver age de gr e e. L et m = 1 2 P n i =1 d i = dn 2 b e the numb er of e dges, and η = 1 n P n i =1 ( d i − d ) 2 . Supp ose that 1 ≪ d ≤ (1 − ε 0 ) n for some ε 0 > 0 . Then, g ( d ) = Q n i =1  n − 1 d i  m 1 / 2  ( n 2 ) m  exp  O (1) − Θ( η 2 /d 2 )  . Pr o of. Deﬁne λ = d/ ( n − 1), N =  n 2  , and note that m = λN . F rom the o verlapping theorems of [ 38 , 42 , 43 ], w e know that g ( d ) ∼ √ 2 exp  1 4 − η 2 4 λ 2 (1 − λ ) 2 n 2   λ λ (1 − λ ) 1 − λ  N n Y i =1  n − 1 d i  . (8) By Stirling’s form ula,  N λN  ∼  λ λ (1 − λ ) 1 − λ  − N  2 π λ (1 − λ ) N  − 1 / 2 , pro vided λ (1 − λ ) N → ∞ . The theorem no w follows from noting that λ (1 − λ ) n = Θ( d ) and λ (1 − λ ) N = Θ( m ) if λ is b ounded aw ay from 1. 8 W e say that a sequence d = ( d 1 , . . . , d n ) is b alanc e d if | d i − d j | ≤ 1 for any 1 ≤ i < j ≤ n . The next observ ation is that g ( d ) is maximized (ov er all sequences with a ﬁxed even sum) when d is balanced. Lemma 10. Fix m ∈  n 2  . The numb er of gr aphs on n vertic es and m e dges with a sp e ciﬁe d de gr e e se quenc e d = ( d 1 , . . . , d n ) (in p articular, P d i = 2 m ) is maximize d when the de gr e e se quenc e is as even as p ossible. In other wor ds, max n g ( d ) : X d i = 2 m o = g ( ˆ d ) , wher e ˆ d is the de gr e e se quenc e, unique up to or der, with only ⌊ P d i /n ⌋ and ⌈ P d i /n ⌉ . Pr o of. Let us ﬁx m ∈  n 2  . F or a con tradiction, supp ose that g ( d ), the num b er of graphs on n vertices with the degree sequence d = ( d 1 , . . . , d n ) satisfying P d i = 2 m , is maximized for a non-balanced sequence d . Without loss of generality , supp ose that d 1 ≤ d 2 − 2. Let d ′ = ( d ′ 1 , . . . , d ′ n ) = ( d 1 + 1 , d 2 − 1 , d 3 , . . . , d n ) with P d ′ i = P d i = 2 m . Consider the follo wing switching: in a graph with the degree sequence d , c ho ose a vertex v suc h that 1 ≁ v and 2 ∼ v , remov e edge { 2 , v } and add { 1 , v } instead (“switch the edges { 2 , v } and { 1 , v } ”) to get a graph with the degree sequence d ′ . Letting g x ( d ) b e the n um b er of graphs where vertices 1 and 2 ha v e exactly x common neighbours, w e get g x ( d )( d 2 − x ) = g x ( d ′ )( d 1 + 1 − x ) < g x ( d ′ )( d 2 − x ) , since d 1 ≤ d 2 − 2. Indeed, there are d 2 − x graphs with the degree sequence d ′ that can b e obtained from a given graph with the degree sequence d and each graph with the degree sequence d ′ can b e obtained from d 1 + 1 − x graphs with the degree sequence d . It follo ws that g ( d ′ ) = P x g x ( d ′ ) > P x g x ( d ) = g ( d ), giving us the desired contradiction. W e also recall the probability b ound on the even t that a uniformly random graph with a given degree sequence con tains a sp eciﬁed set of edges. Claim 11 ([ 41 ]) . L et G n b e a uniformly r andom gr aph on the vertex set [ n ] with a ﬁxe d de gr e e se quenc e ( d 1 , . . . , d n ) . L et H b e a gr aph on the vertex set [ n ] with de gr e e se quenc e ( d ′ 1 , . . . , d ′ n ) such that d ′ i ≤ d i for al l i ∈ [ n ] . L et m = 1 2 P d i and m ′ = 1 2 P d ′ i b e the numb er of e dges in G n and H , r esp e ctively. L et d = max { d 1 , . . . , d n } = o ( m 1 / 2 ) and let m ′ ≤ m/ 2 . Then, P ( H ⊆ G n ) ≤ Q n i =1 d i ( d i − 1) . . . ( d i − d ′ i + 1) 2 m ′ ( m − 2 d 2 )( m − 2 d 2 − 1) . . . ( m − 2 d 2 − m ′ + 1) . W e note that Theorem 11 immediately implies the follo wing. Claim 12. Under the assumptions of The or em 11 , for al l n lar ge enough, P ( H ⊆ G n ) ≤  2 d n  | E ( H ) | . 2.4 Sandwic hing Graphs Consider the binomial random graph G ( n, p ) whic h has v ertex set [ n ] and eac h p otential edge is included indep enden tly at random with probability p ; p = p ( n ) could b e, and usually is, a function 9 of n that tends to zero as n → ∞ . Since the indep endence of the edges allows the use of a wide v ariet y of techniques, G ( n, p ) is t ypically m uch easier to study compared to G n,d . As a result, it is tempting to hop e for a general purp ose “blac k b o x” theorem that is able to translate results b et ween G ( n, p ) and G n,d . In 2004, Kim and V u [ 33 ] formalized this desire in their famous “sandwic h conjecture”. After more than 20 years and a n um b er of important con tributions [ 20 , 23 , 24 , 25 , 34 ], the conjecture w as ﬁnally prov ed [ 12 ]. Theorem 13 (Theorem 1.1 [ 12 ]) . F or e ach ϵ > 0 ther e is some C > 0 such that the fol lowing holds for e ach d ≥ C log n . Ther e is a c oupling ( G ∗ , G , G ∗ ) of r andom gr aphs such that G ∗ ∼ G ( n, (1 − ϵ ) d/n ) , G ∼ G n,d , G ∗ ∼ G ( n, (1 + ϵ ) d/n ) , and whp G ∗ ⊂ G ⊂ G ∗ . 2.5 Expansion Prop erties W e will use the expansion prop erties of random d -regular graphs that follow from their eigen v alues. The adjacency matrix A = A ( G ) of a giv en d -regular graph G on n vertices, is an n × n real symmetric matrix. Thus, the matrix A has n real eigen v alues whic h w e denote by d = λ 1 ≥ λ 2 ≥ · · · ≥ λ n . It is known that several structural prop erties of a d -regular graph are reﬂected in its sp ectrum. Since we fo cus on expansion prop erties, w e are particularly interested in the following quan tit y: λ = λ ( G ) := max {| λ 2 | , | λ n |} . The num b er of edges e ( A, B ) b etw een tw o sets A and B in a random d -regular graph on n v ertices is exp ected to b e close to d | A || B | /n . (Note that A ∩ B does not hav e to b e empt y; in general, e ( A, B ) is deﬁned to b e the num b er of edges b et ween A \ B to B plus t wice the num b er of edges that contain only vertices of A ∩ B .) A small λ (that is, a large sp ectral gap) implies that the deviation is small. The following b ound is very conv enien t. Theorem 14 (Expander Mixing Lemma [ 3 , 35 ]) . L et G b e a d -r e gular gr aph. Then for any two sets of vertic es A, B ⊆ V ( G ) , the numb er e ( A, B ) of e dges of G with one endp oint in A and another endp oint in B satisﬁes     e ( A, B ) − d | A || B | n     ≤ λ p | A || B | . W e will apply the Expander Mixing Lemma together with an asymptotic bound on λ for random regular graphs. It was ﬁrst established b y F riedman [ 21 ] for constan t d ≥ 3, conﬁrming the conjecture of Alon [ 2 ]. The case of d → ∞ was then conjectured by V u [ 49 ]. After a series of imp ortan t contributions [ 4 , 16 , 18 , 36 , 47 ], it was resolved for all d = o ( n ) by Bauersc hmidt, Huang, Knowles, and Y au [ 11 ] and Sarid [ 46 ], and then for d = Θ( n ) by He [ 27 ]. Theorem 15 ([ 11 , 27 , 46 ]) . L et 3 ≤ d ≤ n/ 2 and G n ∼ G n,d . Then, whp λ ( G n ) ≤ (2 + o (1)) p d (1 − d/n ) . W e will also require a ﬁner expansion result ensuring that, for ev ery set, its size remains con- cen trated after sev eral rounds of expansion. Lemma 16. L et c > 0 b e smal l enough and d 0 ∈ N b e lar ge enough (indep endent of c ). L et d 0 ≤ d ≤ n/ 2 and G n ∼ G n,d . Then, the fol lowing holds whp: for every set U of size u = | U | ≤ cn d and for every p ositive inte ger r such that ud ( d − 1) r − 1 ≤ cn , | S r ( U ) | ≥ (1 − 100 c − 4 ln d/d ) ud ( d − 1) r − 1 . The pro of of this lemma is techni cally inv olved. T o av oid interrupting the ﬂo w of the pap er, we presen t it separately in Section 7 . 10 3 Pro of of Theorem 4 : Dense Case Here we prov e the following. Theorem 17. L et ε ∈ (0 , 1 / 2) , n 1 / 2+ ε ≤ d = d ( n ) ≤ n/ 2 , and let G n ∼ G n,d . L et k ∈ N b e an arbitr ary c onstant. Then, the fol lowing holds whp: for every non-trivial p artition [ n ] = V 1 ⊔ V 2 of the vertex set of G n , after two r ounds of CR, ther e exists a p artition [ n ] = V 1 ⊔ . . . ⊔ V k such that for any i ∈ [ k ] , n/ 3 k ≤ | V i | ≤ 3 n/k and V i is a union of some c olour classes. Theorem 17 follo ws easily from the follo wing tw o claims. Claim 18. L et ε ∈ (0 , 1 / 2) , n 1 / 2+ ε ≤ d = d ( n ) ≤ n/ 2 , and let G n ∼ G n,d . F or every C > 1 , the fol lowing pr op erty holds whp: for every non-trivial p artition [ n ] = V 1 ⊔ V 2 with min {| V 1 | , | V 2 |} < C n/d , after one r ound of CR, ther e exists a p artition [ n ] = U 1 ⊔ U 2 with min {| U 1 | , | U 2 |} ≥ C n/d such that U 1 and U 2 ar e unions of some c olour classes (in other wor ds, U 1 ⊔ U 2 is a c o arsening of the p artition asso ciate d with r esulting c olour classes). Claim 19. L et ε ∈ (0 , 1 / 2) , n 1 / 2+ ε ≤ d = d ( n ) ≤ n/ 2 , and let G n ∼ G n,d . F or any δ ∈ (0 , 1] , ther e exists C = C ( δ ) > 0 such that the fol lowing pr op erty holds whp: for every non-trivial p artition [ n ] = V 1 ⊔ V 2 with min {| V 1 | , | V 2 |} ≥ C n/d , ther e is no c olour class of size mor e than δ n after one r ound of CR. Before we prov e these tw o claims, let us sho w ho w they imply Theorem 17 . Pr o of of The or em 17 . Since we aim for the statement that holds whp, w e may assume that the statemen ts in Theorem 18 and in Theorem 19 hold deterministically . Fix any k ∈ N , and let δ = 1 / (3 k ). Let C = C ( δ ) be the large enough constan t implied by Theorem 19 . Consider an y non-trivial partition [ n ] = V 1 ⊔ V 2 . If min {| V 1 | , | V 2 |} < C n/d , then after one round of CR (and coarsening), we get a partition into t w o colour classes in which one of the colour classes has size at least C n/d but at most n/ 2 (b y Theorem 18 ). After another round of CR, all colour classes hav e size at most δ n = n/ (3 k ) (by Theorem 19 ). If min {| V 1 | , | V 2 |} ≥ C n/d , then we get the ab o ve prop ert y after a single round of CR. T o get the desired partition into k parts, each of size at least n/ (3 k ), one can iteratively merge an y tw o colour classes of size at most δ n until there is at most one such class remaining. After p ossibly merging this last class (if it exists) with any other arbitrarily chosen class, we get at least k = 1 / (3 δ ) classes (but at most 3 k of them), each of size at least δ n = n/ 3 k but at most 3 δ n . Finally , if there are more than k classes, one can arbitrarily merge some triples of them (and, p erhaps, one pair) to get exactly k classes, each of size at most 9 δ n = 3 n/k . This ﬁnishes the pro of of the theorem. Theorem 18 has a short and easy pro of so let us start with it. Pr o of of The or em 18 . Consider any partition [ n ] = V ⊔ ([ n ] \ V ) with 1 ≤ | V | < C n/d . After one round of CR, [ n ] \ V gets reﬁned into colour classes eac h of which consists of vertices with the same n um b er of neighbours in V . F or i ∈ { 0 } ∪ N , let W i ⊆ [ n ] \ V be the set of v ertices in [ n ] \ V with precisely i neigh b ours in V . If | V | ≤ n/ 2 d , then w e can simply take U 1 = S i ≥ 1 W i (v ertices in [ n ] \ V with at least one neigh b our in V ) and U 2 = [ n ] \ U 1 . T rivially , since the graph is d -regular, | U 1 | ≤ d | V | ≤ d · ( n/ 2 d ) = n/ 2. On the other hand, since d ≥ n 1 / 2+ ε , | U 1 | ≥ d − | V | ≥ d − n/ 2 d = (1 + o (1)) d ≥ C n/d . Hence, the partition [ n ] = U 1 ⊔ U 2 satisﬁes the desired prop ert y . 11 T o deal with sets V satisfying n/ 2 d < | V | < C n/d , w e will tak e U 1 = S i ≥ t W i for some threshold v alue t that will dep end on | V | . The v alue of t will b e carefully selected so that U 1 has size at most n/ 2 but at least εn for some ε > 0. T o bound the size of U 1 , w e will use the “sandwic h theorem” ( Theorem 13 ). Let p − = 0 . 9 d/n and p + = 1 . 1 d/n . The “sandwic h theorem” implies that there exists a coupling b etw een tw o copies of the binomial random graph and a random d -regular graph suc h that whp G ( n, p − ) ⊆ G n,d ⊆ G ( n, p + ). Since we aim for a statement that holds whp, w e ma y use this coupling to get the desired b ounds. F or a given set V ⊆ [ n ] of size v with n/ 2 d < v < C n/d , let X + b e the n um b er of vertices in [ n ] \ V in G ( n, p + ) that are adjacent to at least t := 4 dv /n vertices in V , noting that X + ≥ | U 1 | . F or a given vertex y ∈ [ n ] \ V , the num b er of neigh b ours of y in V is Y ∼ Bin( v , p + ) with the exp ectation equal to λ = 1 . 1 dv /n = Θ(1); note that 0 . 55 < λ < 1 . 1 C . Using ( 1 ) we get that q + := P ( Y ≥ t ) = P  Y ≥ 4 λ 1 . 1  ≤ exp  − (4 / 1 . 1 − 1) 2 2(1 + (4 / 1 . 1 − 1) / 3) λ  < e − 1 . 84 λ < e − 1 . On the other hand, since λ = Θ(1), Y tends to a Poisson random v ariable with parameter λ in distribution. As a result, q + > ξ + for some univ ersal constan t ξ + > 0 and so q + = Θ(1). Coming back to X + , let us note that X + ∼ Bin( n − v , q + ) with E X + = ( n − v ) q + = (1 + o (1)) nq + = Θ( n ) (here is the place where w e use the fact that q + > ξ + > 0). It follows immediately from Chernoﬀ ’s bound ( 1 ) that P  X + ≥ n 2  ≤ exp ( − Θ( n )) . By the union b ound ov er X n/ 2 d ξ − > 0. As a result, E X − = ( n − v ) q − = (1 + o (1)) nq − = Θ( n ) and w e get concentration with exp onen tially small failure probability . The union b ound ov er all choices of V giv es us the desired universal lo w er b ound of, sa y , nξ − / 2 ≥ C n/d for X − in G ( n, p − ) for all V that holds whp. This ﬁnishes the pro of of the claim, since G ( n, p − ) ⊆ G n,d whp so this low er b ound also holds whp in G n,d . The pro of of Theorem 19 is more complex. Let us start with the following simple observ ation. Lemma 20. L et ε ∈ (0 , 1 / 2) , n 1 / 2+ ε ≤ d = d ( n ) ≤ n/ 2 , and let G n ∼ G n,d . F or any ﬁxe d δ > 0 , the fol lowing pr op erty holds whp: for any U ⊆ [ n ] of size u = | U | > δ n and any V ⊆ [ n ] \ U of size v = | V | ≥ 500 n/dδ , the numb er of e dges e ( U, V ) b etwe en U and V satisﬁes the fol lowing b ounds 0 . 8 uv · d n − 1 ≤ e ( U, V ) ≤ 1 . 2 uv · d n − 1 . Pr o of. Let p − = 0 . 9 d/ ( n − 1) and p + = 1 . 1 d/ ( n − 1). The “sandwic h theorem” ( Theorem 13 ) implies that there exists a coupling b et ween tw o copies of the binomial random graph and a random d - 12 regular graph such that whp G ( n, p − ) ⊆ G n,d ⊆ G ( n, p + ). Since we aim for a statement that holds whp, we may use this coupling to get the desired b ounds. Fix any U ⊆ [ n ] of size u = | U | > δ n and any V ⊆ [ n ] \ U of size v = | V | ≥ 500 n/dδ . In particular, note that uv d > 500 n 2 . Let X − b e the n umber of edges b et ween U and V in G ( n, p − ). Clearly , X − ∼ Bin( uv , p − ) with E X − = 0 . 9 uv d/ ( n − 1). It follows immediately from Chernoﬀ ’s b ound ( 2 ) that P  X − ≤ 0 . 8 uv · d n − 1  ≤ exp  − 0 . 1 2 2 · 0 . 9 uv · d n − 1  ≤ exp  − uv d 250 n  ≤ e − 2 n . By the union b ound ov er at most (2 n ) 2 c hoices for U and V , we get the desired low er b ound for e ( U, V ) (for an y pair U , V ) in G ( n, p − ) that holds whp. Since G ( n, p − ) ⊆ G n,d whp, this property also holds whp in G n,d . A symmetric argument can be used to get the desired upp er bound. Indeed, if X + is the num ber of edges b et w een U and V in G ( n, p + ), then E X + = 1 . 1 uv d/ ( n − 1) and Chernoﬀ ’s b ound ( 1 ) gives us that P  X + ≥ 1 . 2 uv · d n − 1  ≤ exp  − 0 . 1 2 2(1 . 1 + 0 . 1 / 3) uv · d n − 1  ≤ exp  − uv d 250 n  . The upp er bound holds whp for G ( n, p + ) and so it holds also whp for G n,d since whp G n,d ⊆ G ( n, p + ). Remark 21. Theorem 20 also follows from the Expander Mixing Lemma and Theorem 15 . No w, we can mo v e to the pro of of Theorem 19 . Pr o of of The or em 19 . Fix any δ ∈ (0 , 1] and let C = C ( δ ) be a large enough constan t that will b e sp eciﬁed later. In particular, we will assume that C ≥ 500 /δ so that w e may apply Theorem 20 . Supp ose that there exists a partition [ n ] = V ⊔ ([ n ] \ V ) with C n/d ≤ | V | ≤ n/ 2 such that after one round of CR there exists a colour class U of size more than δ n . Note that this implies that ev ery v ertex in U has the same n um b er of neigh b ours in V (hence every v ertex in U also has the same num b er of neigh b ours in [ n ] \ V ). If U ⊆ V , then it will be con v enien t to concen trate on the n um b er of neigh b ours in [ n ] \ V ⊆ [ n ] \ U but if U ⊆ [ n ] \ V , then we will concen trate on the n umber of neigh b ours in V ⊆ [ n ] \ U . Our goal is to estimate the probabilit y of the w eaker but necessary prop ert y that there exists a pair of sets ( U, V ) suc h that V ⊆ [ n ] \ U , | V | ≥ C n/d , | U | > δ n , and ev ery vertex in U has the same n um b er of neighbours in V . Fix V ⊆ [ n ] and U ⊆ [ n ] \ V such that v := | V | ≥ C n/d and u := | U | > δ n . Note that, in particular, v < (1 − δ ) n . F or eac h non-negativ e in teger k ≤ v , deﬁne E k ( U, V ) to b e the even t that ev ery vertex in U has exactly k neighbours in V , in which case the num b er of edges b et w een U and V is exactly uk . By Theorem 20 , since w e aim for a statement that holds whp, we may assume that the n umber of edges b etw een U and V is at least 0 . 8 uv d/ ( n − 1) and at most 1 . 2 uv d/ ( n − 1). Hence, we may restrict to considering k such that 0 . 8 v · d n − 1 ≤ k ≤ 1 . 2 v · d n − 1 . (9) No w, note that the exp ected num b er of edges induced b y [ n ] \ V is  n − v 2  · d n − 1 . W e will sho w 13 that it is highly unlikely that the actual num ber deviates substantially from it. Let m − = 0 . 9  n − v 2  · d n − 1 and m + = 1 . 1  n − v 2  · d n − 1 . By Theorem 9 and the Stirling’s formula ( s ! = (1 + o (1)) √ 2 π s ( s/e ) s ), letting d := ( d, . . . , d ) ∈ Z n , the num b er of d -regular graphs on [ n ] can b e estimated as follo ws: g ( d ) = Θ    n − 1 d  n √ dn  ( n 2 ) dn/ 2    = (1 + o (1)) r n 2 π d ( n − d ) ·  n − 1 e  n − 1  d e  d  n − 1 − d e  n − 1 − d ! n (10) ·    Θ(1)  n ( n − 1) / 2 e  n ( n − 1) / 2  dn/ 2 e  dn/ 2  n ( n − 1 − d ) / 2 e  n ( n − 1 − d ) / 2    − 1 = Θ(1) ·   n 2  dn/ 2   (1 + o (1)) r n 2 π d ( n − d )  n =   n 2  dn/ 2  d − Θ( n ) . Hence, the probability that the n um b er of edges induced by [ n ] \ V is at most m − or at least m + can b e upp er b ounded by X m ≤ m − m ≥ m +  ( n − v 2 ) m  ( n 2 ) − ( n − v 2 ) dn/ 2 − m  g ( d ) = n Θ( n ) X m ≤ m − m ≥ m +  ( n − v 2 ) m  ( n 2 ) − ( n − v 2 ) dn/ 2 − m   ( n 2 ) dn/ 2  = n Θ( n ) · P ( η ≤ m − or η ≥ m + ) , where η is the hypergeometric random v ariable with parameters  n 2  ,  n − v 2  , and dn/ 2. Clearly , E η = dn 2 ·  n − v 2   n 2  =  n − v 2  d n − 1 = Θ( dn ) . By Chernoﬀ ’s bound for the hypergeometric distribution (see, ( 1 ), ( 2 ), and Theorem 6 ), P ( η ≤ m − or η ≥ m + ) = P ( | η − E η | ≥ 0 . 1 E η ) = exp ( − Ω ( E η )) = exp( − Ω( dn )) . Similarly , if | V | ≥ n 3 / 4 , then the exp ected n umber of edges induced b y V is  v 2  · d n − 1 = Θ( v 2 d/n ) and we get that with probability exp( − Ω( dv 2 /n )) = exp( − Ω( dn 1 / 2 )), the num b er of edges induced b y V is at most 0 . 9  v 2  · d n − 1 or at least 1 . 1  v 2  · d n − 1 . It remains to concentrate on the case when the n um b er of edges induced by [ n ] \ V is at least m − but at most m + , that is, when the av erage degree of the graph induced b y [ n ] \ V is at least 0 . 9( n − 1 − v ) · d n − 1 but at most 1 . 1( n − 1 − v ) · d n − 1 . Let us ﬁrst deal with the case when | V | ≥ n 3 / 4 so w e may additionally assume that the av erage degree of the graph induced b y V is at least 0 . 9( v − 1) · d n − 1 but at most 1 . 1( v − 1) · d n − 1 . By Theorem 9 and Theorem 10 , P ( E k ( U, V )) = e − Ω( dn ) + e − Ω( dn 1 / 2 ) + O X D h ( k , u, v , D ) ! , (11) 14 where D denotes the n umber of edges b et w een V and [ n ] \ ( V ∪ U ), and h ( k , u, v , D ) :=  v k  u  ( n − v − u ) v D  v − 1 d − ( D + k u ) /v  v  n − 1 − v d − ( D + k u ) / ( n − v )  n − v  ( v 2 ) ( dv − ku − D ) / 2  ( n − v 2 ) ( d ( n − v ) − ku − D ) / 2  g ( d ) . Indeed, there are at most  v k  u w a ys to place edges b et ween U and V , and at most  ( n − v − u ) v D  w a ys to place edges betw een V and [ n ] \ ( V ∪ U ). (Note that these v alues are trivial upper b ounds but not the exact ones as some c hoices create v ertices of degree more than d .) It remains to estimate the n um b er of graphs induced b y the set [ n ] \ V and the num b er of graphs induced by the set V . Imp ortan tly , once other edges are ﬁxed, these graphs ha v e a ﬁxed degree distribution. In particular, the av erage degree of the graphs induced b y [ n ] \ V is precisely f ( D ) := d − ( D + k u ) / ( n − v ). Similarly , the a verage degree of the graphs induced b y V is ˜ f ( D ) := d − ( D + ku ) /v . Hence, we ma y use Theorem 9 and Theorem 10 to get upp er b ounds for the num ber of such graphs, which in turn imply ( 11 ). (Let us p oin t out that f ( D ) and ˜ f ( D ) are not necessarily in tegers. How ever, to k eep the notation simple, we write  n − 1 − v f ( D )  n − v instead of the pro duct of n − v terms, each of them b eing  n − 1 − v ⌊ f ( D ) ⌋  or  n − 1 − v ⌈ f ( D ) ⌉  .) Finally , since the a v erage degree of the graph induced b y [ n ] \ V and the one induced b y V are restricted, D satisﬁes the requirements 0 . 9( n − 1 − v ) · d n − 1 ≤ f ( D ) ≤ 1 . 1( n − 1 − v ) · d n − 1 , (12) 0 . 9( v − 1) · d n − 1 ≤ ˜ f ( D ) ≤ 1 . 1( v − 1) · d n − 1 . (13) There are three binomials in the numerator of h ( k , u, v , D ) that are raised to p o wers that are functions of n . W e need to tak e adv antage of them using Theorem 8 (see ( 6 )). By ( 12 ) and recalling v < (1 − δ ) n ,  n − 1 − v f ( D )  n − v ≤  (1 + o (1))( n − 1 − v ) f ( D )( n − 1 − v − f ( D ))  ( n − v − 1) / 2  ( n − v )( n − v − 1) d ( n − v ) − k u − D  ≤  1 + o (1) 0 . 9 · δ · d · (1 − 1 . 1 · 0 . 5)  ( n − v − 1) / 2  ( n − v )( n − v − 1) d ( n − v ) − k u − D  ≤  3 δ d  ( n − v − 1) / 2  ( n − v )( n − v − 1) d ( n − v ) − k u − D  . (14) Similarly , by ( 13 ),  v − 1 ˜ f ( D )  v ≤  (1 + o (1))( v − 1) ˜ f ( D )( v − 1 − ˜ f ( D ))  ( v − 1) / 2  v ( v − 1) dv − k u − D  ≤  1 + o (1) 0 . 9 · ( v d/n ) · (1 − 1 . 1 · 0 . 5)  ( v − 1) / 2  v ( v − 1) dv − k u − D  ≤  3 n v d  ( v − 1) / 2  v ( v − 1) dv − k u − D  . (15) 15 Finally , by ( 9 ),  v k  u ≤  v k ( v − k )  ( u − 1) / 2  v u k u  ≤  1 + o (1) 0 . 8 v d/n (1 − 1 . 2 · 0 . 5)  ( u − 1) / 2  v u k u  ≤  4 n dv  ( u − 1) / 2  v u k u  . (16) F or future reference, let us highligh t that ( 15 ) only holds when v ≥ n 3 / 4 , whilst the other tw o b ounds ( 14 ) and ( 16 ) hold in general. Substituting in these three bounds, w e get h ( k , u, v , D ) = O   n 3 / 2  4 n dv  u/ 2  v u ku  ( n − v − u ) v D   3 n v d  v / 2  v ( v − 1) dv − ku − D   3 δ d  ( n − v ) / 2  ( n − v )( n − v − 1) d ( n − v ) − ku − D   ( v 2 ) ( dv − ku − D ) / 2  ( n − v 2 ) ( d ( n − v ) − ku − D ) / 2  g ( d )   . No w, by Theorem 8 (see ( 7 )), h ( k , u, v , D ) = O   n 7 / 2  4 n dv  u/ 2  v u ku  ( n − v − u ) v D   3 n v d  v / 2  ( v 2 ) ( dv − ku − D ) / 2   3 δ d  ( n − v ) / 2  ( n − v 2 ) ( d ( n − v ) − ku − D ) / 2  g ( d )   . By Theorem 8 (see ( 4 )), w e can collect all binomial co eﬃcien ts together to get h ( k , u, v , D ) = O   n 7 / 2  4 n dv  u/ 2  v u +( n − v − u ) v + ( v 2 ) + ( n − v 2 ) ku + D +( dv − ku − D ) / 2+( d ( n − v ) − k u − D ) / 2   3 n v d  v / 2  3 δ d  ( n − v ) / 2 g ( d )   = O   n 7 / 2  4 n dv  u/ 2  ( n 2 ) dn/ 2   3 n v d  v / 2  3 δ d  ( n − v ) / 2 g ( d )   . Using ( 10 ) w e get that g ( d ) ≥   n 2  dn/ 2  ((2 + o (1)) π d ) − n/ 2 ≥   n 2  dn/ 2  (7 d ) − n/ 2 , and so h ( k , u, v , D ) = O n 7 / 2  4 n dv  u/ 2  3 n v d  v / 2  3 δ d  ( n − v ) / 2 (7 d ) n/ 2 ! = O n 7 / 2  4 n dv  u/ 2  δ n v  v / 2  21 δ  n/ 2 ! . No w, let h ( v ) := ( δ n/v ) v / 2 and note that ∂ ∂ v log h = 1 2 log δ n ev . Hence, h ( v ) is maximized for v = δ n/e 16 and we get that for any p ositiv e in teger v ,  δ n v  v / 2 ≤ max v h ( v ) = exp  δ n 2 e  ≤ 2 n/ 2 , since exp(1 /e ) ≈ 1 . 445 ≤ 2 and δ ≤ 1. Since u > δ n and v ≥ C n/d , it follows that h ( k , u, v , D ) = O n 7 / 2  4 n dv  u/ 2  42 δ  n/ 2 ! = O   n 7 / 2  4 C  δ · 42 δ ! n/ 2   = O  5 − n  , pro vided that C is large enough so that, say ,  4 C  δ · 42 δ ≤ 42 / 1200 = 0 . 035 < 0 . 04 = 1 / 5 2 . Since δ is ﬁxed, this condition can be easily satisﬁed and w e may now ﬁnally deﬁne the constant C : C = C ( δ ) := max  4 ( δ / 1200) 1 /δ , 500 δ  . W e conclude that if | V | ≥ n 3 / 4 , then P ( E k ( U, V )) = e − Ω( dn ) + e − Ω( dn 1 / 2 ) + O X D h ( k , u, v , D ) ! = O  n 2 5 − n  . If v < n 3 / 4 then, as men tioned earlier, we do not get the term  3 n v d  ( v − 1) / 2 in the estimation of h ( k , u, v , D ) (see ( 15 )). How ever, for v < n 3 / 4 , this term do es not help us m uch anyw ay:  3 n v d  ( v − 1) / 2 = exp( − Ω( v log v )) ≥ exp( − n 4 / 5 ). Hence, regardless of the size of V , P ( E k ( U, V )) = O  n 2 exp( n 4 / 5 ) 5 − n  . Finally , by the union bound, P ( ∃ k , U, V E k ( U, V )) ≤ n · 2 n · 2 n · O  n 2 exp( n 4 / 5 ) 5 − n  = o (1) , whic h ﬁnishes the pro of of the theorem. 4 Pro of of Theorem 4 : Sparse Case Sparser graphs clearly require more rounds of CR. Consider any d -regular graph with diameter D and let u and v b e any tw o vertices at distance D from each other. CR run on the initial partition V 1 = { v } and V 2 = [ n ] \ { v } requires at least D − 1 rounds to con verge to a discrete colouring. Indeed, after D − 2 rounds there are at least t wo v ertices at distance at least D − 1 from v that are still of the same colour. In this section, we pro v e the follo wing. Theorem 22. L et d 0 b e lar ge enough, let d 0 ≤ d = o ( n ) , and let G n ∼ G n,d . L et k ∈ N b e an arbitr ary c onstant. Then, the fol lowing holds whp: for every non-trivial p artition [ n ] = V 1 ⊔ V 2 of the vertex set of G n , after at most diam( G n ) + 2 r ounds of CR, ther e exists a p artition [ n ] = V 1 ⊔ . . . ⊔ V k such that for any i ∈ [ k ] , n/ 3 k ≤ | V i | ≤ 3 n/k and V i is a union of some c olour classes. 17 W e start from auxiliary an ti-concentration results that will be used to pro v e Theorem 22 in Sec- tion 4.2 . 4.1 An ti-concen tration Results Lemma 23. L et ℓ b e a lar ge enough c onstant, and let d 0 = d 0 ( ℓ ) b e another lar ge enough c onstant. L et d 0 ≤ d = o ( n ) and G n ∼ G n,d . Then, the fol lowing pr op erty holds whp: for every set U of size | U | ∈ [ nℓ d , n 2 ] and every non-ne gative inte ger s , the numb er of vertic es in [ n ] \ U that have exactly s neighb ours in U is at most 10 n/ ln ℓ . Pr o of. Set t = 10 n/ ln ℓ . The Expander Mixing Lemma and Theorem 15 imply that whp, for every set U of size m ∈ [ nℓ/d, n/ 2] and ev ery set V ⊂ [ n ] \ U of size t , • the n um b er of edges betw een U and V is at most min { dm/ 4 , 3 td/ 4 } , • the n um b er of edges induced b y V is at most 0 . 01 dt , • the n um b er of edges betw een U and [ n ] \ ( U ∪ V ) is at least md/ 3, • the n um b er of edges betw een [ n ] \ ( U ∪ V ) and U ∪ V is at most 2 d ( n − m − t ) / 3. Due to the ﬁrst bullet p oin t, it suﬃces to prov e the lemma for s suc h that 10 sn/ ln ℓ ≤ dm/ 4 and st ≤ 3 td/ 4. Therefore, w e ﬁx an in teger s ∈  0 , min  3 d 4 , dm ln ℓ 40 n  . (17) W e also ﬁx a set U of size m ∈ [ nℓ d , n 2 ] and a set V ⊂ [ n ] \ U of size t . Let us estimate the probabilit y that every vertex from V has exactly s neigh b ours in U . Let us order the vertices in V arbitrarily: x 1 , x 2 , . . . , x t . Exp ose the edges in V . Let d − d i b e the degree of x i in G n [ V ]. Let V ′ ⊂ V be the set of all vertices with d i ≥ 0 . 8 d . Due to the second bullet p oin t ab o ve, we may assume that P d i ≤ 0 . 02 dt and therefore, | V ′ | ≥ 0 . 9 t . Denote t ′ = | V ′ | . Without loss of generalit y we assume that V ′ = { x 1 , . . . , x t ′ } . Let E b e the even t that every x i from V ′ has s neigh b ours in U . Let h in ≥ d ( n − m − t ) / 6 and h out ≥ md/ 3 b e integers suc h that P ( E ∧ {| E ( G n [[ n ] \ ( U ∪ V )]) | = h in , | E ( G n [ U × ([ n ] \ ( U ∪ V ))] | = h out } ) is maxim um . Let Σ 0 b e the set of all d -regular graphs G on [ n ] satisfying E and such that G [ V ] = G n [ V ] and G [[ n ] \ ( U ∪ V )] and G [ U × ([ n ] \ ( U ∪ V ))] hav e exactly h in and h out edges, resp ectiv ely . The follo wing claim completes the pro of of Theorem 23 . Claim 24. P ( G n ∈ Σ 0 | G n [ V ]) ≤ ℓ − t ′ / 5 . Indeed, b y the union b ound ov er U, V and the num b er of edges in the graphs G [[ n ] \ ( U ∪ V )] and G [ U × ([ n ] \ ( U ∪ V ))], w e get that probabilit y that there exist sets U, V such that E holds is at most o (1) + ( nd ) 2 n/ 2 X m = nℓ/d  n m  n t  ℓ − t ′ / 5 t ′ ≤ 0 . 9 t ≤ o (1) + n 4 2 n  en tℓ 9 / 50  t ≤ o (1) + n 4 2 n  ln ℓ ℓ 9 / 50  10 n/ ln ℓ ≤ o (1) + n 4 2 n e − n = o (1) . 18 Pr o of of The or em 24 . Assume the opp osite: P ( G n ∈ Σ 0 ) > ℓ − t ′ / 5 . (18) Let X b e the set of d -regular graphs G on [ n ] such that there exists a v ertex v with at least dm/ℓ edges betw een N ( v ) and U . Then, due to Theorem 12 , P ( G n ∈ X ) ≤ n  dm md/ℓ   2 d n  md/ℓ ≤ n  2 eℓd n  md/ℓ = o ( e − n ) . (19) Fix an integer δ such that | δ | ≤ ℓ 0 . 3 . Let Σ 1 ( δ ) b e the set of d -regular graphs G on [ n ] such that • G [ V ] = G n [ V ], • eac h v ertex x 2 , . . . , x t ′ has exactly s neigh b ours in U , • x 1 has s + δ neigh b ours in U , and • there are h in + δ edges in G [[ n ] \ ( V ∪ U )] and h out − δ edges in G [ U × ([ n ] \ ( U ∪ V ))] . W e shall pro v e that       [ | δ |≤ ℓ 0 . 3 Σ 1 ( δ )       ≥ ℓ 0 . 2 | Σ 0 | . (20) Let us ﬁrst compare sizes of Σ 1 (1) and Σ 0 . T ake G ∈ Σ 0 and consider a tuple of v ertices ( y , u, v ) suc h that • y / ∈ U ∪ V , u ∈ U , v / ∈ U ∪ V , • and { x 1 , y } , { u, v } ∈ E ( G ), { x 1 , u } , { y , v } / ∈ E ( G ). If we switch { x 1 , y } , { u, v } 7→ { x 1 , u } , { y , v } , (21) w e get a graph from Σ 1 (1). F or every G ∈ Σ 0 \ X the num ber of forwar d switchings (i.e., tuples ( y , u, v )) equals ( d 1 − s )( h out − µ ), where µ ∈ [0 , 2 dm/ℓ ] since, ﬁrst, G / ∈ X and so the n umber of edges b et w een N ( y ) and U is at most dm/ℓ , and second, the num b er of edges that touch N ( x 1 ) ∩ U is at most sd = O ( d 2 m/n ) < dm/ℓ due to ( 17 ). On the other hand, for every graph G ∈ Σ 1 (1), the n um b er of b ackwar d switchings equals ( s + 1)( h in + 1 − ν ), where ν ∈ [0 , 2 d 2 ]. W e get | Σ 0 | ( d 1 − s )( h out − µ ) − |X ∩ Σ 0 | O ( d 2 n ) = | Σ 1 (1) | ( s + 1)( h in + 1 − ν ) . In a similar w a y , w e compare the sizes of Σ 1 ( − 1) and Σ 0 : F or G ∈ Σ 0 , w e take a tuple of vertices ( y , u, v ) suc h that • y ∈ U , u, v / ∈ U ∪ V , • and { x 1 , y } , { u, v } ∈ E ( G ), { x 1 , u } , { y , v } / ∈ E ( G ), and switch as in ( 21 ). W e get | Σ 0 | s ( h in − ν ′ ) = | Σ 1 ( − 1) | ( d 1 − s + 1)( h out + 1 − µ ′ ) − |X ∩ Σ 1 ( − 1) | O ( d 2 n ) , where ν ′ ∈ [0 , 2 d 2 ], and µ ′ ∈ [0 , 2 dm/ℓ ] . 19 If ( d 1 − s ) h out > sh in , then | Σ 1 (1) | | Σ 0 | > 1 − O (1 /ℓ ) − O (1 /s ) − O  d 2 n · |X | | Σ 0 |  = 1 − O (1 /ℓ ) − O (1 /s ) due to ( 19 ) and the assumption ( 18 ) and the lo w er b ounds on h out and h in . Otherwise, | Σ 1 ( − 1) | | Σ 0 | > 1 − O (1 /ℓ ) . (22) First we assume ( d 1 − s ) h out > sh in . If s < ℓ 0 . 7 , then ( d 1 − s ) h out = Ω( ℓ 0 . 3 sh in ) since h out = Θ( md ) and h in = Θ( dn ). W e conclude that | Σ 1 (1) | / | Σ 0 | = Ω( ℓ 0 . 3 ) > ℓ 0 . 2 , completing the pro of of ( 20 ). Let s ≥ ℓ 0 . 7 . F or δ = 1 , . . . , ℓ 0 . 3 − 1, w e deﬁne a switc hing operation that maps Σ 1 ( δ ) to Σ 1 ( δ + 1) in exactly the same wa y . W e get | Σ 1 ( δ ) | ( d 1 − s − δ )( h out − δ − µ δ ) − |X ∩ Σ 1 ( δ ) | O ( d 2 n ) = | Σ 1 ( δ + 1) | ( s + δ + 1)( h in + 1 + δ − ν δ ) , for some ν δ ∈ [0 , 2 d 2 ] and µ δ ∈ [0 , 2 dm/ℓ ]. W e then get | Σ 1 ( δ + 1) | | Σ 1 ( δ ) | > 1 − O ( δ /s ) − O (1 /ℓ ) = 1 − O ( δ /ℓ 0 . 7 ) , implying | Σ 1 ( δ + 1) | | Σ 0 | = | Σ 1 (1) | | Σ 0 | · δ Y j =1 | Σ 1 ( j + 1) | | Σ 1 ( j ) | > δ +1 Y j =1 (1 − O ( j /ℓ 0 . 7 )) = 1 − O ( δ 2 /ℓ 0 . 7 ) = 1 − O ( ℓ − 0 . 1 ) . Consequen tly , we can deduce       [ | δ |≤ ℓ 0 . 3 Σ 1 ( δ )       ≥       ℓ 0 . 3 [ δ =1 Σ 1 ( δ )       ≥ (1 − O ( ℓ − 0 . 1 )) ℓ 0 . 3 | Σ 0 | > ℓ 0 . 2 | Σ 0 | , as needed. No w, let ( d 1 − s ) h out ≤ sh in . In particular, s = Ω( ℓ ). Recall ( 22 ). Similarly , for δ = 1 , . . . , ℓ 0 . 3 − 1, we deﬁne a switching op eration that maps Σ 1 ( − δ ) to Σ 1 ( − δ − 1) and get | Σ 1 ( − δ ) | ( s − δ )( h in − δ − ν ′ δ ) = | Σ 1 ( − δ − 1) | ( d 1 − s + δ + 1)( h out + 1 + δ − µ ′ δ ) − |X ∩ Σ 1 ( − δ − 1) | O ( d 2 n ) , for some ν ′ δ ∈ [0 , 2 d 2 ] and µ ′ δ ∈ [0 , 2 dm/ℓ ]. W e then get | Σ 1 ( − δ − 1) | | Σ 1 ( − δ ) | > 1 − O ( δ /s ) − O (1 /ℓ ) = 1 − O ( δ /ℓ ) , implying | Σ 1 ( − δ − 1) | | Σ 0 | > δ +1 Y j =1 (1 − O ( j /ℓ )) = 1 − O ( δ 2 /ℓ ) = 1 − O ( ℓ − 0 . 4 ) . 20 Finally , this giv es       [ | δ |≤ ℓ 0 . 3 Σ 1 ( δ )       ≥       ℓ 0 . 3 [ δ =1 Σ 1 ( − δ )       ≥ (1 − O ( ℓ − 0 . 4 )) ℓ 0 . 3 | Σ 0 | > ℓ 0 . 2 | Σ 0 | , as needed. Let Σ 1 := S | δ |≤ ℓ 0 . 3 Σ 1 ( δ ). W e no w deﬁne sets Σ 2 , Σ 3 , . . . , Σ t ′ as follows: for | δ | ≤ ℓ 0 . 3 , let Σ i ( δ ) b e the set of d -regular graphs G on [ n ] such that • G [ V ] = G n [ V ], • eac h v ertex x i +1 , . . . , x t ′ has exactly s neigh b ours in U , • x i has s + δ neigh b ours in U , • for ev ery j ∈ [ i − 1], | N U ( x j ) − s | ≤ ℓ 0 . 3 , • there are h in + P i − 1 j =1 ( | N U ( x j ) | − s ) + δ edges in G [[ n ] \ ( U ∪ V )] and h out − P i − 1 j =1 ( | N U ( x j ) | − s ) − δ edges in G [ U × ([ n ] \ ( U ∪ V ))] . Let Σ i := S | δ |≤ ℓ 0 . 3 Σ i ( δ ). The pro of that | Σ i +1 | ≥ ℓ 0 . 2 | Σ i | is iden tical to that of ( 20 ) since i − 1 X j =1 ( | N U ( x j ) | − s ) ≤ tℓ 0 . 3 = O ( h out / ( ℓ 0 . 7 ln ℓ )) , and so it do es not con tribute signiﬁcantly to ν δ , µ δ , ν ′ δ , and µ ′ δ . Therefore, | Σ 0 | ≤ ℓ − 0 . 2 | Σ 1 | ≤ ℓ − 0 . 4 | Σ 2 | ≤ . . . ≤ ℓ − 0 . 2 t ′ | Σ t ′ | — a con tradiction whic h completes the pro of of the claim. The pro of of the claim w as the last piece of the puzzle in establishing the lemma. Lemma 25. L et ℓ b e a lar ge enough c onstant, and let d 0 = d 0 ( ℓ ) b e another lar ge enough c onstant. L et d 0 ≤ d = o ( n ) and G n ∼ G n,d . Then, the fol lowing pr op erty holds whp: for every set U of size | U | ∈ [ n 2 d , nℓ d ] and every inte ger s such that 1 ≤ s ≤ ℓ , ther e ar e at most 0 . 999 n vertic es that have exactly s neighb ours in U . Pr o of. Let ε = 0 . 001. Let E b e the ev en t that, for any disjoin t sets V and W of size n/ 4 and εn , respec tiv ely , the num b er of edges in V ∪ W that touch V is at most dn/ 30. By the Expander Mixing Lemma and Theorem 15 the ev en t E holds whp. Fix a set U of size m ∈ [ n 2 d , ℓn d ] and a set V = { x 1 , . . . , x t } ⊂ [ n ] \ U of size n (1 − ε ). Divide V = V ′ ⊔ V ′′ , where V ′ consists of the ﬁrst n/ 4 vertices. Exp ose edges inside V ′ and b etw een V ′ and [ n ] \ ( U ∪ V ) and assume that E := E ( G n [ V ′ ]) ∪ E ( G n [ V ′ × ([ n ] \ ( U ∪ V ))]) has size at most dn/ 30. Let ˜ V ′ ⊂ V ′ b e the set of vertices that b elong to at most d/ 2 edges in E . Clearly , | ˜ V ′ | ≥ n/ 12. Without loss of generalit y , we assume ˜ V ′ = { x 1 , . . . , x t ′ } , where t ′ ≥ n/ 12. Note that eac h x i , i ∈ [ t ′ ], has at least d/ 2 − s neigh b ours in V ′′ . Let Σ 0 b e the set of d -regular graphs G on [ n ] such that E ( G [ V ′ ]) ∪ E ( G [ V ′ × ([ n ] \ ( U ∪ V ))]) = E (23) 21 and each v ertex in V has exactly s neigh b ours in U . Let Σ 1 b e the set of d -regular graphs G on [ n ] such that ( 23 ) is satisﬁed and • eac h vertex x 2 , . . . , x t has exactly s neigh b ours in U , except for some x i ∈ V ′′ , whereas x i has s − 1 neigbhours in U , • x 1 has s + 1 neigh b ours in U . W e shall prov e that | Σ 1 | ≥ | Σ 0 | / 3. T ake G ∈ Σ 0 and consider a tuple of vertices ( y , u, v ) such that • y ∈ V ′′ , u ∈ U , v ∈ V ′′ , • and { x 1 , y } , { u, v } ∈ E ( G ), { x 1 , u } , { y , v } / ∈ E ( G ). If we switch { x 1 , y } , { u, v } 7→ { x 1 , u } , { y , v } , (24) w e get a graph from Σ 1 . F or ev ery G ∈ Σ 0 the num b er of forwar d switchings is at least ( d/ 2 − s )(((3 / 4 − ε ) n − d ) s − sd ): there are at least d/ 2 − s choices of y , at least (3 / 4 − ε ) n − d choices of v ∈ V ′′ that is not adjacen t to y , and, therefore, at least ((3 / 4 − ε ) n − d ) s c hoices of the edge { u, v } where u ∈ U . It then remains to subtract at most sd edges { u, v } where u ∈ U is a neighbour of x 1 . On the other hand, for every graph G ∈ Σ 1 , the num b er of b ackwar d switchings is at most ( s + 1)(3 n/ 4) d , with a ro om for improv ement since the choice of v = x i is actually unique. W e get | Σ 0 | ( d/ 2 − s )(((3 / 4 − ε ) n − d ) s − sd ) ≤ | Σ 1 | ( s + 1)(3 n/ 4) d implying | Σ 1 | ≥ | Σ 0 | / 3, as desired. W e now let Σ 2 b e the set of d -regular graphs G on [ n ] such that ( 23 ) is satisﬁed and • eac h v ertex x 3 , . . . , x t has exactly s neighbours in U , except for some x i 1 , x i 2 ∈ V ′′ that ha ve s − 1 neighbours in U , • | N U ( x 1 ) | ∈ [ s, s + 1], • x 2 has s + 1 neigh b ours in U . T ake G ∈ Σ 0 ∪ Σ 1 and consider a tuple of v ertices ( y , u, v ) suc h that • y ∈ V ′′ , u ∈ U , v ∈ V ′′ , and | N U ( v ) | = s , • { x 2 , y } , { u, v } ∈ E ( G ), { x 2 , u } , { y , v } / ∈ E ( G ). If w e switc h as in ( 24 ) with x 2 instead of x 1 , then we get a graph from Σ 2 . As ab o v e, for every G ∈ Σ 0 ∪ Σ 1 the n umber of forwar d switchings is at least ( d/ 2 − s )(((3 / 4 − ε ) n − d − 1) s − sd ). On the other hand, for every graph G ∈ Σ 1 , the num b er of b ackwar d switchings is at most ( s + 1)(3 n/ 4) d , as ab o v e. W e get | Σ 0 ∪ Σ 1 | ( d/ 2 − s )(((3 / 4 − ε ) n − d − 1) s − sd ) ≤ | Σ 2 | ( s + 1)(3 n/ 4) d implying | Σ 2 | ≥ | Σ 0 ∪ Σ 1 | / 3, as w ell. Similarly , we deﬁne Σ 3 , . . . , Σ n/ 12 . F or the i -th set Σ i , we get that | Σ 0 ∪ . . . ∪ Σ i − 1 | ( d/ 2 − s )(((3 / 4 − ε ) n − d − ( i − 1)) s − sd ) ≤ | Σ i | ( s + 1)(3 n/ 4) d, 22 implying | Σ i | ≥ | Σ 0 ∪ . . . ∪ Σ i − 1 | / 3 for all i ≥ 1. It is easy to see by induction that | Σ i | ≥ (4 / 3) i − 1 | Σ 0 | / 3 for all i ≥ 1. Thus, we get | Σ n/ 12 | > (4 / 3) n/ 12 − 1 | Σ 0 | / 3 , implying P ( G n ∈ Σ 0 | E ( G n [ V ′ ]) ∪ E ( G n [ V ′ × ([ n ] \ ( U ∪ V ))]) = E ) < 4 · (4 / 3) − n/ 12 . Denote E n ( U, V ) := E ( G n [ V ′ ]) ∪ E ( G n [ V ′ × ([ n ] \ ( U ∪ V ))]). The union b ound o v er U and V giv es us that P ( ¬E ) +  n εn  n ℓn/d  X E : | E |≤ dn/ 15 P ( G n ∈ Σ 0 | E n ( U, V ) = E ) · P ( E n ( U, V ) = E ) = o (1) + e ( ε ln( e/ε )+ o d (1)) n (4 / 3) − n/ 12 = o (1) , whic h completes the pro of of the lemma. 4.2 Pro of of Theorem 22 With Theorems 16 , 23 and 25 at hand, w e can easily prov e Theorem 22 . Pr o of of The or em 22 . Fix a large enough 5 k ∈ N , and let ℓ = e 30 k , so that the conclusions of The- orems 23 and 25 hold. Let c > 0 be a small enough constant as in Theorem 16 . Let d 0 = d 0 ( ℓ ) b e a large enough constant as in Theorems 16 , 23 and 25 . Moreov er, we will adjust constants c or d 0 , if needed, for some of the claims b elo w to hold. Since we aim for the statemen t that holds whp, we ma y assume that the statements in Theorems 16 , 23 and 25 hold deterministically . Consider any non-trivial partition [ n ] = V 1 ⊔ V 2 . Our goal is to sho w that after at most diam( G n ) + 2 man y rounds of CR, we get a partition into colour classes in which all colour classes ha v e size at most n/ 3 k . T o get the desired partition into k parts, eac h of size at least n/ 3 k but at most 3 n/k , one can iterativ ely merge colour classes as w e did in the pro of of Theorem 17 . Let u = min {| V 1 | , | V 2 |} and let U be a colour class of size u . Supp ose ﬁrst that u < cn/d . Let r b e the largest in teger suc h that ud ( d − 1) r − 1 ≤ cn . W e may adjust c and d , if needed, to mak e sure that 1 − 100 c − 4 ln d/d ≥ 1 / 2, which, in particular, implies that c ≤ 1 / 200. It follows from Theorem 23 that | S r ( U ) | ≥ (1 − 100 c − 4 ln d/d ) ud ( d − 1) r − 1 > cn 2 d , and clearly | S r ( U ) | ≤ ud ( d − 1) r − 1 ≤ cn ≤ n/ 2. After r ≤ diam( G n ) − 1 rounds of CR, S r ( U ) is a union of some colour classes. W e ma y marge them together at this point and contin ue the pro cess from there. Supp ose no w that U is a colour class of size u = | U | ∈ [ cn 2 d , n 2 d ]. Let U ′ b e an arbitrary subset of U of size cn 2 d . On the one hand, trivially , | S 1 ( U ) | ≤ d | U | ≤ n/ 2. On the other hand, it follo ws from Theorem 23 that | S 1 ( U ′ ) | ≥ | U ′ | d/ 2 = cn/ 4 which implies that | S 1 ( U ) | ≥ | S 1 ( U ′ ) | − | U | ≥ cn/ 4 − n/ (2 d ) ≥ nℓ/d , provided that d is large enough. Since S 1 ( U ) is a union of some colour classes, we may merge them in to one large colour class and con tin ue from there. Supp ose this time that U is a colour class of size u = | U | ∈ [ n 2 d , nℓ d ]. W e ma y adjust d , if needed, to make sure u ≤ nℓ d ≤ n 2( ℓ +1) . After one round of CR, [ n ] \ U is partitioned in to sets W i 5 Clearly , if the statemen t of the theorem is true for large k , then it is also true for all k . 23 ( i ∈ N ∪ { 0 } ); set W i consists of v ertices with exactly i neigh b ours in U . Let A = S i ≤ ℓ W i and let B = S i ≥ ℓ +1 W i . Clearly , | U | + | A | + | B | = n . Note that, on the one hand, the num b er of edges b et ween U and its complement is at least | B | ( ℓ + 1). On the other hand, it is trivially at most | U | d ≤ nℓ . W e conclude that | B | ≤ ℓ ℓ + 1 n =  1 − 1 ℓ + 1  n, and so | A | = n − | B | − | U | ≥ n ℓ + 1 − u ≥ n 2( ℓ + 1) . Our goal is to show that one can alwa ys merge some sets W i together to get a colour class of size at most n/ 2 but at least n 2( ℓ +1) , which is at least nℓ d , provided that d is large enough. T o that end, w e will consider a few cases. If | A | ≤ n/ 2, then we can simply take the en tire set A for the desired colour class. If | A | > n/ 2 but | A | ≤ (1 − 1 / ( ℓ + 1)) n , then w e may take the en tire set B since | B | = n − | A | − | U | ≥ n/ ( ℓ + 1) − u ≥ n/ (2( ℓ + 1)) and, trivially , | B | = n − | A | − | U | < n/ 2. It remains to concen trate on the case when | A | ≥ (1 − 1 / ( ℓ + 1)) n . Supp ose ﬁrst that | W i | ≥ n/ 2 for some 0 ≤ i ≤ ℓ . It follows from Theorem 25 that | W i | ≤ 0 . 999 n . Then, we can take A \ W i for the desired colour class since, trivially , | A \ W i | ≤ n − | W i | ≤ n/ 2 and | A \ W i | ≥ | A | − | W i | ≥ 0 . 001 n − n ℓ + 1 ≥ nℓ d , pro vided that d is large enough. If n/ 4 ≤ | W i | < n/ 2 for some 0 ≤ i ≤ ℓ , then we may simply tak e W i as our colour class. Supp ose then that | W i | < n/ 4 for all 0 ≤ i ≤ ℓ . Then, we may start with set A and remov e W i ’s, one by one, and at some p oin t we get a set of size at most n/ 2 but at least n/ 4. Finally , supp ose that U is a colour class of size u = | U | ∈ [ nℓ d , n 2 ]. It follo ws immediately from Theorem 23 that after one round of CR, the complement of U is partitioned into sets of size at most 10 n/ ln ℓ = n/ 3 k . W e can group some of them together to get a colour class of size at least n/ 3 k ≥ nℓ/d but at most 2 n/ 3 k ≤ n/ 2 to mak e sure that after one more round U is also partitioned in to sets of size at most n/ 3 k . This completes the pro of of the theorem. 5 Pro of of Theorem 5 : Dense Case Here, we prov e the following. Theorem 26. L et ε > 0 , let n 1 / 2+ ε ≤ d ≤ n/ 2 , and let G n ∼ G n,d . Ther e exists some universal lar ge c onstant k ∈ N such that the fol lowing holds whp: for every p artition [ n ] = V 1 ⊔ . . . ⊔ V k of the vertex set of G n such that for any i ∈ [ k ] , n/ 3 k ≤ | V i | ≤ 3 n/k , after thr e e r ounds of CR, ther e ar e only singleton c olour classes. W e start from an auxiliary lemma. Lemma 27. L et ε > 0 , n − 1 / 2+ ε ≤ d = d ( n ) ≤ n/ 2 , and let G n ∼ G n,d . F or every p air of vertic es u, v ∈ [ n ] , let M u,v ⊂ N ( u ) and M ′ u,v ⊂ N ( v ) \ N ( u ) b e sets of size ⌊ n/ (20 d ) ⌋ chosen uniformly at r andom. Then the fol lowing events hold whp for any p air of vertic es u, v in G n : 1. | N ( u ) ∩ N ( v ) | < 2 d/ 3 ; 2. | N ( u, v ) | < 4 n/ 5 ; 24 3. | N ( M ′ u,v ) \ ( N ( { u, v } ) ∪ N ( M u,v )) | > n/ 25 whenever d ≤ n/ 20 . Pr o of. Let p − = 0 . 9 d/ ( n − 1) and p + = 1 . 1 d/ ( n − 1). Theorem 13 implies that there exists a coupling b et w een tw o copies of the binomial random graph and a random d -regular graph suc h that whp G ( n, p − ) ⊆ G n,d ⊆ G ( n, p + ). Since w e aim for a statement that holds whp, we may use this coupling to obtain the ﬁrst tw o b ounds. It is suﬃcien t to show that for a ﬁxed pair of v ertices u, v an y of the ﬁrst t w o even ts from the assertion of the claim fails in both G ( n, p − ) and G ( n, p + ) with probability o ( n 2 ). First we consider the probabilit y that N ( u ) ∩ N ( v ) is large. In b oth binomial random graphs the num b er of shared neighbours is dominated by Bin( n − 1 , p 2 + ). Since ( n − 1) p 2 + ≤ 0 . 61 d , the Chernoﬀ b ound ( 1 ) implies that | N ( u ) ∩ N ( v ) | ≥ 2 d/ 3 in G ( n, p + ) with probabilit y exp( − Ω( d 2 /n )). Similarly N ( u ) ∪ N ( v ) is dominated by Bin( n − 1 , 2 p + (1 − p + / 2)). Since 2( n − 1) p + (1 − p + / 2) ≤ 0 . 798 n , the Chernoﬀ b ound ( 1 ) implies that | N ( u ) ∪ N ( v ) | ≥ 4 n/ 5 in G ( n, p + ) with probabilit y exp( − Ω( d )). Next, let d < n/ 20. By the same argument, w e get that whp ev ery pair of v ertices has at most 1 . 2 d 2 n common neighbours. Therefore, | N ( M ′ u,v ) \ ( N ( { u, v } ) ∪ N ( M u,v )) | > l n 20 d m  d −  n 10 d + 1  1 . 2 d 2 n  > n 25 , whic h ﬁnishes the pro of of the lemma. Pr o of of The or em 26 . Due to the Expander Mixing Lemma and Theorem 15 , whp b etw een any set N of size Ω( d ) and an y set W of size Ω( n ), there are (1 ± o (1)) | N || W | d n edges. W e denote the in tersection of this ev en t with the even t from the assertion of Theorem 27 by E . Supp ose that k is as large as needed, and ﬁx an y partition [ n ] = V 1 ⊔ . . . ⊔ V k suc h that eac h part has size in the range [ n/ 3 k , 3 n/k ] as in the statement of the theorem. F or each i ∈ [ k ], deﬁne d i : [ n ] → Z so that d i ( w ) = | V i ∩ N ( w ) | . Finally , for each vertex u ∈ [ n ], we interpret c i ( u ) as the colour of u after i rounds of CR. F or our goal, it suﬃces to show that whp no tw o vertices hav e the same v alue of c 3 ( · ). W e proceed as follows. Fix a pair of v ertices u, v ∈ [ n ] and exp ose the neighbourho ods of u and v . Note that c 3 ( u ) = c 3 ( v ) if and only if there exists a bijection b : N ( v ) 7→ N ( u ) suc h that for an y w ∈ N ( v ) w e hav e c 2 ( b ( w )) = c 2 ( w ). Fix such a bijection (in d ! w ays). Deﬁne N ′ := N ( v ) \ ( N ( u ) ∪ { u } ). Cho ose arbitrarily a set M ′ ⊂ N ′ of ⌊ n/ (20 d ) ⌋ v ertices from N ′ , and let M = b − 1 ( M ′ ) ⊂ N ( U ). Exp ose all edges that touch M ∪ N ′ . Let M ′′ = N ′ ∪ N ( M ′ ) \ ( N ( u ) ∪ N ( M ) ∪ { u, v } ) . W e extend the bijection b to an injection b : N ( V ) ∪ N ( M ) 7→ N ( U ) ∪ N ( M ′ ) such that, for ev ery w ∈ M and every w ′ ∈ N ( w ), we get c 1 ( b ( w ′ )) = c 1 ( w ′ ) and b ( w ′ ) ∈ N ( b ( w )). The n umber of w a ys to deﬁne suc h an extension is at most ( d !) n/ 20 d . Let W := [ n ] \ ( { u, v } ∪ N ( u ) ∪ N ( v ) ∪ N ( M ) ∪ N ( M ′ )), and W i := W ∩ V i . After exp osing every edge except those b et w een M ′′ and W , we can determine the v alues of d i ( w ) for every w ∈ N ( M ) (as every neigh b our of every v ertex in N ( M ) has b een exp osed). Therefore, the injection b also iden tiﬁes d i ( w ) for every i ∈ [ k ] and ev ery w ∈ M ′′ . Let S denote the num b er of pairs ( i, w ) with 1 ≤ i ≤ k and w ∈ M ′′ suc h that | W i | ≥ n/ (30 k ) and | W i | d/ (2 n ) ≤ | N ( w ) ∩ W i | ≤ 3 | W i | d/ (2 n ). 25 The num b er of w ays to choose the remaining neighbours of the vertices in M ′′ is k Y i =1 Y w ∈ M ′′  | W i | d i ( w )  ≤  960 k d  S/ 2  | W || M ′′ | dn/ 2 − m  , where m is the num b er of exp osed edges. Indeed for any p ositiv e integers a 1 , a 2 , b 1 , b 2 with b 1 ≥ b 2 and b i ≤ 3 a i / 4 for i = 1 , 2 we ha ve by Theorem 8 that  a 1 b 1  a 2 b 2  ≤ 2 3 s ( b 1 + b 2 )( a 1 + a 2 ) a 1 a 2 ( a 1 + a 2 ) b 1 ( a 1 − b 1 ) b 2 ( a 2 − b 2 )  a 1 + a 2 b 1 + b 2  ≤ 8 3 r b 1 + b 2 b 1 b 2  a 1 + a 2 b 1 + b 2  ≤ 4 √ b 2  a 1 + a 2 b 1 + b 2  . Let us show that the ev ent E implies S ≥ k n/ 1100. Indeed, this even t implies that | W | ≥ n/ 5 (if d > n/ 20, then W = [ n ] \ ( N ( { u, v } ) ∪ { u, v } ) and has size at least n/ 5 by the second assertion of Theorem 27 ; if d ≤ n/ 20, then | W | ≥ n − 2 d − 2( n/ (20 d )) d > n/ 5). Therefore, there are at least n/ 6 v ertices in the union of W i suc h that | W i | ≥ n/ (30 k ). Thus, there are at least ( n/ 6) / (3 n/k ) = k / 18 suc h W i . Fix suc h a W i . Since E holds, any subset ˜ N ⊂ M ′′ of size Ω( n ) sends (1 ± o (1)) | ˜ N || W i | edges to W i . Moreo v er, | M ′′ | ≥ n/ 60. Indeed, if d > n/ 20, then M ′′ = N ′ whic h has size at least d/ 3 > n/ 60 by the ﬁrst assertion of Theorem 27 ; if d ≤ n/ 20, then | M ′′ | > n/ 25 by the third assertion. The even t E also implies that num b er of v ertices in M ′′ that ha v e less than | W i | d 2 n or more than | W i | 3 d 2 n edges in W i , is o ( n ). So, indeed S > (1 − o (1))( k / 18)( n/ 60) > k n/ 1100. Using ( 10 ) and letting g ( d ) = ( d, . . . , d ) ∈ Z n , the probability that there exists u, v ∈ [ n ] and a partition V 1 ⊔ . . . ⊔ V k suc h that c ( u ) = c ( v ) is then at most P ( ¬E ) + 1 g ( d ) n 2 2 kn d ! X N ′ ⊆ [ n ] X W ⊆ [ n ] \ N ′ dn/ 2 X m =0   n 2  − | N ′ || W | m   960 k d  S/ 2  | W || N ′ | dn/ 2 − m  = o (1) + d Θ( n )  960 k d  kn/ 2200 = o (1) , when k is suﬃcien tly large. This completes the pro of of Theorem 26 . 6 Pro of of Theorem 5 : Sparse Case Here we prov e the following. Theorem 28. Ther e exists a universal c onstant k such that the fol lowing holds. L et d 0 = d 0 ( k ) b e lar ge enough, let d 0 ≤ d ≤ n 10 / 17 , and let G n ∼ G n,d . Then whp: for every p artition [ n ] = V 1 ⊔ . . . ⊔ V k of the vertex set of G n such that for any i ∈ [ k ] , n/ 3 k ≤ | V i | ≤ 3 n/k , after diam( G n ) + 1 r ounds of CR, ther e ar e only singleton c olour classes. W e will use t w o direct corollaries of Theorem 16 from Section 4 in this pro of. W e ﬁrst state these in Section 6.1 , and then pro v e Theorem 28 in Section 6.2 . 6.1 Sizes of Balls As in Theorem 16 , let c > 0 b e small enough and d 0 b e large enough. Let d 0 ≤ d = o ( n ) and let G n ∼ G n,d . The following t w o lemmas are direct corollaries of Theorem 16 . 26 Lemma 29. Whp, for every r such that d ( d − 1) r − 1 ≤ cn , the fol lowing holds • for every vertex u , | S r ( u ) | ≥  1 − 100 c − 4 ln d d  d ( d − 1) r − 1 ; • for every p air of vertic es u  = v , | S r ( v ) \ B r ( u ) | ≥ | S r ( { u, v } ) | − | B r ( u ) | ≥  1 − 200 c − 8 ln d d − 1 d − 2  d ( d − 1) r − 1 . Pr o of. The ﬁrst assertion is just Theorem 16 applied with a singleton U = { u } . The second follo ws from | U | = 2 in Theorem 16 together with the basic b ound | B r ( u ) | ≤ P i ≤ r − 2 d ( d − 1) i . Lemma 30. Whp • for every set U of size cn d , ther e ar e at le ast (1 − 4 ln d/d − 100 c ) cn vertic es that have a neighb our in U ; • for any two disjoint sets U, V of size cn/d , the numb er of vertic es that have neighb ours b oth in U and in V is at most | N ( U ) | / 10 . Pr o of. The ﬁrst assertion follows immediately from Theorem 16 applied with r = 1. The second assertion follo ws as well since whp for an y tw o disjoin t sets U and V , the num b er of vertices that ha v e neighbours in b oth sets is at most | N ( U ) | + | N ( V ) | − | N ( U ∪ V ) | ≤ 2 d ( cn/d ) − (1 − 4 ln d/d − 100 c )2 cn = (4 ln d/d + 100 c )2 cn ≤ 1 10 (1 − 4 ln d/d − 100 c ) cn ≤ | N ( U ) | / 10 . This ﬁnishes the pro of of the lemma. 6.2 Colour Reﬁnemen t Run on a V ertex-coloured Random Graph Let d b e large enough. In what follo ws we assume that prop erties from Theorem 29 and Theorem 30 hold in G n deterministically . Fix a partition [ n ] = V 1 ⊔ . . . ⊔ V k as in the statement of the theorem. Assign to every vertex x the colour C 0 ( x ) that equals the index of the set V i to which x b elongs. Let D b e the diameter of G n . Consider the output C t of t := D + 1 rounds of CR at the coloured graph. W e wan t to pro v e that C ( u )  = C ( v ) for an y tw o diﬀerent vertices u, v ∈ [ n ]. Fix t w o v ertices u  = v . Assume C t ( u ) = C t ( v ). Then, for ev ery neighbour x of v , there exists a neighbour y of u suc h that C t − 1 ( x ) = C t − 1 ( y ). More generally , w e hav e the follo wing. Claim 31. L et r ∈ [ t ] . F or every vertex a and every vertex b such that C t − r ( a ) = C t − r ( b ) , and every neighb our x of a , ther e exists a neighb our y or b such that C t − r − 1 ( x ) = C t − r − 1 ( y ) . Let r = ⌊ log d − 1 ( εn ) ⌋ , where ε > 0 is a small enough constant. Due to the prop ert y from the 27 conclusion of Theorem 29 , we hav e that | S r ( u ) \ B r ( v ) | ≥  1 − 4 d − 1 d − 1 − 2 ln n − O ( ε )  d ( d − 1) r − 1 > d ( d − 1) 2 εn  1 − 4 d − 1 d − 1 − 2 ln n − O ( ε )  > ε 2 d · n. Due to Theorem 31 , for every vertex x ∈ S 1 ( u ) \ B 1 ( v ), there exists a vertex f ( x ) ∈ B 1 ( v ) such that C t − 1 ( x ) = C t − 1 ( f ( x )). Next, for every vertex x ∈ S 2 ( u ) \ B 2 ( v ), let π ( x ) ∈ S 1 ( u ) \ B 1 ( v ) b e one of its “parents”. W e ha v e that C t − 1 ( π ( x )) = C t − 1 ( f ( π ( x )). Therefore, b y Theorem 31 , there exists f ( x ) ∈ N ( f ( π ( x ))) ⊂ B 2 ( v ) suc h that C t − 2 ( x ) = C t − 2 ( f ( x )). W e then deﬁne f : B r ( u ) \ B r ( v ) → B r ( v ) b y induction: for every 2 ≤ i ≤ r , assuming that f has b een deﬁned on B i − 1 ( u ) \ B i − 1 ( v ), and for ev ery x ∈ S i ( u ) \ B i ( v ), ﬁnd its “paren t” π ( x ) ∈ S i − 1 ( u ) \ B i − 1 ( v ) and tak e f ( x ) ∈ N ( f ( π ( x )) suc h that C t − i ( x ) = C t − i ( f ( x )). T ake U ⊂ S r ( u ) \ B r ( v ) of size εn 2 d and let U ′ := f ( U ) ⊂ B r ( v ). W e hav e | U ′ | ≤ | U | . Without loss of generality , w e assume | U ′ | = | U | (otherwise, we can extend U ′ arbitrarily to keep the t w o sets disjoint, and the argumen t b elo w will still work). Note that B r ( u ) = B r − 1 ( u ) ∪ N ( S r − 1 ( u )), that | S r − 1 ( u ) | ≤ d ( d − 1) r − 2 ≤ d ( d − 1) 2 εn < 1 . 1 ε n d , and that U and S r − 1 are disjoint. The same facts hold for B r ( v ). In particular, | S r − 1 ( u ) ∪ S r − 1 ( v ) | < 3 ε n d . Therefore, by the conclusion of Theorem 30 , we hav e that | N ( U ) \ ( N ( U ′ ) ∪ B r ( u ) ∪ B r ( v )) | > | N ( U ) | − 4 10 | N ( U ) | − | B r − 1 ( u ) | − | B r − 1 ( v ) | > 6 10 (1 − (ln 40) / 10 − Θ( ε )) 1 2 εn − 2 d d − 2 ( d − 1) r − 1 > 0 . 18 · εn − 2 dεn ( d − 2)( d − 1) > 0 . 1 · εn. Let N b e a subset of N ( U ) \ ( N ( U ′ ) ∪ B r ( u ) ∪ B r ( v )) of size εn/ 10. W e then extend f to N : Each vertex x ∈ N has f ( x ) ∈ N ( U ′ ). Note that the set X := [ n ] \ ( B r ( u ) ∪ B r ( v ) ∪ N ∪ f ( N )) has size at least n − 2 d ( d − 1) r − 1 − εn 2 d · (2 d ) > n − 2 d d − 1 ε · n − ε · n ≥ n (1 − 4 ε ) . The set X is partitioned in to k sets X = V ′ 1 ⊔ . . . ⊔ V ′ k so that n ( 1 3 k − ε ) ≤ | V ′ i | ≤ n · 3 k . Due to Theorem 12 , whp an y set of size at most εn/ 3 induces at most 1 100 εdn edges:  n εn/ 3  ε 2 n 2 / 18 1 100 εdn   2 d n  1 100 εdn ≤  3 e ε  100 / 3 (100 eε/ 9) d ! 1 100 εn = o (1) 28 since d is large and ε is small enough. In particular, w e may assume that there there are at most 1 100 εdn edges b et ween N and N ∪ f ( N ) ∪ U . Since N do es not hav e any other neighbours in B r ( u ) ∪ B r ( v ), we get that there exists a subset N 0 ⊂ N of size εn 50 suc h that eac h vertex in this set sends at least 3 4 d edges to X . Note that, for any v ertex x ∈ N 0 , the equality C t − r − 1 ( x ) = C t − r − 1 ( f ( x )) implies C 1 ( x ) = C 1 ( f ( x )). Therefore, as so on as the sets B r ( u ) , B r ( v ) are exp osed, the set U is c hosen, the sets N ( U ) , N ( U ′ ) , N ( N ( U ′ )) are exp osed, and the set N 0 is c hosen, there should exist a function f deﬁned as abov e, that identiﬁes the v alues of | N V ′ j ( x ) | for ev ery x ∈ N 0 and j ∈ [ k ]. Therefore, w e run the following exploration pro cess of the random graph. First, we exp ose B r ( u ) , B r ( v ) and then choose U ⊂ S r ( u ) \ B r ( v ) of size εn 2 d arbitrarily . W e then exp ose N ( U ), N ( U ′ ), and N ( N ( U ′ )). W e c ho ose f on ( B r ( u ) \ B r ( v )) ∪ N in at most d 2 εn w a ys, since | ( B r ( u ) \ B r ( v )) ∪ N | ≤ | B r ( u ) | + | N ( U ) | ≤ d d − 2 ( d − 1) r + εn 2 ≤ d d − 2 εn + εn 2 < 2 εn. Finally , we choose any set N 0 ⊂ N of size εn 50 suc h that eac h v ertex in this set sends at least 3 4 d edges to X . By the Expander Mixing Lemma and Theorem 15 , whp b et w een any tw o disjoint sets of size at least n/ (4 k ) and n/ 2, there are at least dn/ (10 k ) edges, and ev ery set of size at least n/ 2 induces at least dn/ 10 edges. Recall that ev ery x ∈ N 0 has a prescrib ed num b er of neigh b ours g j ( x ) in the set V ′ j . By the Expander Mixing Lemma and Theorem 15 , whp the num b er of edges b et ween an y tw o disjoint sets U and V of sizes Θ( n ) equals | U || V | d (1 ± ε ) /n . Therefore, for ev ery set V ′ j , there exists a subset N ′ j ⊂ N 0 of size εn/ 100 suc h that ev ery x ∈ N ′ j has g j ( x ) ∈ [ d/ (10 k ) , 10 d/k ] . Let us estimate the probabilit y that for ev ery j ∈ [ k ], ev ery v ertex from N ′ j has g j ( x ) neigh b ours in V ′ j . F or ev ery j , w e order arbitrarily the v ertices in N ′ j : x j 1 , . . . , x j t , where t = εn/ 100 . Let E b e the ev ent that, for ev ery j ∈ [ k ], ev ery x j i has g j ( x j i ) neigh b ours in V ′ j . Let h j in ≥ dn/ 10 and h j out ≥ dn/ (10 k ) b e integers such that P   E ∧ k ^ j =1 n | E ( G n [ X \ V ′ j ] | = h j in , | E ( G n [ V ′ j × ( X \ V ′ j )] | = h j out o   is maximum . Let Σ 0 b e the set of all d -regular graphs G on [ n ] satisfying E and suc h that, for all j ∈ [ k ], G [ X \ V ′ j ] and G [ V ′ j × ( X \ V ′ j )] ha ve exactly h j in and h j out edges, resp ectiv ely 6 . The follo wing claim completes the pro of of Theorem 23 . Claim 32. P ( G n ∈ Σ 0 ) ≤ ( k /d ) εnk/ 500 . 6 Sets X and V ′ j dep end on G : given a graph G , we exp ose the balls around u and v , which identify these sets. In what follows, we will perform switc hing op erations on G that preserv e the exp osed balls and, therefore, sets X and V ′ j . 29 Indeed, Theorem 32 implies that P ( E ) ≤ ( dn ) 2 k ( k /d ) εnk/ 500 . Therefore, by the union bound P ( C t ( u ) = C t ( v )) ≤ d 2 εn ( dn ) 2 k ( k /d ) εnk/ 500 = e − Ω( kn ) when k is large enough and d ≫ k . The union b ound ov er the choice of partition V 1 ⊔ . . . ⊔ V k and o v er all pairs of distinct v ertices u, v completes the proof of Theorem 28 . The pro of of Theorem 32 can basically b e obtained by cop ying the pro of of Theorem 24 verbatim. Nev ertheless, we present the pro of b elo w in nearly complete detail. Pr o of of The or em 32 . Assume the opp osite: P ( G n ∈ Σ 0 ) > ( k /d ) εnk/ 500 . Let X b e the set of d - regular graphs G on [ n ] suc h that there exists a v ertex x and a set V of size n/ (3 k ) with at least n ( d/k ) 0 . 3 edges b et w een N ( x ) and V . Then, due to Theorem 12 , P ( G n ∈ X ) ≤ n  n n/ (3 k )  3 dn/k n ( d/k ) 0 . 3   2 d n  n ( d/k ) 0 . 3 ≤ n  n n/ (3 k )   3 e ( d/k ) 0 . 7 · 2 d n  n ( d/k ) 0 . 3 = o  e − n ( d/k ) 0 . 3  , (25) since d ≤ n 10 / 17 and k is large enough. In the same w a y as in the pro of of Theorem 24 , w e deﬁne sets Σ j ℓ , j ∈ [ k ], ℓ ∈ [ t ]: for | δ | ≤ ( d/k ) 0 . 3 , we deﬁne Σ j ℓ ( δ ) as the set of d -regular graphs G on [ n ] such that • for ev ery i ≥ j + 1, each vertex x ∈ N ′ i has exactly g i ( x ) neighbours in V ′ i , • for ev ery ℓ + 1 ≤ i ≤ t , the v ertex x j i has exactly g j ( x j i ) neighbours in V ′ j , • x j ℓ has g j ( x j ℓ ) + δ neighbours in V ′ j , • for every i ≥ j − 1, the num b er of neighbours of every vertex x ∈ N ′ i in V ′ i satisﬁes | N V ′ i ( x ) − g i ( x ) | ≤ ( d/k ) 0 . 3 ; • for ev ery i ∈ [ ℓ − 1], | N V ′ j ( x j i ) − g j ( x j i ) | ≤ ( d/k ) 0 . 3 , • for every i ∈ [ k ], the num b er of edges in G [ X \ V ′ i ] and G [ V ′ i × ( X \ V ′ i )], denoted by h i in ( j, ℓ ) and h i out ( j, ℓ ), respectively , satisfy:   h i in ( ℓ, j ) − h i in   ≤ ( d/k ) 0 . 3 (( j − 1) t + ℓ ) ,   h i out ( ℓ, j ) − h i out   ≤ ( d/k ) 0 . 3 (( j − 1) t + ℓ ) . Set Σ j ℓ := S | δ |≤ ( d/k ) 0 . 3 Σ j ℓ ( δ ). The crucial observ ation is that, at every step, the b ound on the diﬀerence of the n um b er of edges in X \ V ′ i and V ′ i × ( X \ V ′ i ) with h i in and h i out is at most ( d/k ) 0 . 3 k t = O  min j { h j out , h j in } · k 1 . 7 /d 0 . 7  . In order to get a con tradiction with our initial assumption, we prov e that • | Σ 0 | ≤ ( d/k ) − 0 . 2 | Σ 1 1 | , • for ev ery j ∈ [ k ] and ℓ ∈ [ t − 1], | Σ j ℓ | ≤ ( d/k ) − 0 . 2 | Σ j ℓ +1 | , • for ev ery j ∈ [ k − 1], | Σ j t | ≤ ( d/k ) − 0 . 2 | Σ j +1 1 | . 30 W e prov e this using switchings. F or instance, in order to pro v e | Σ j ℓ | ≤ ( d/k ) − 0 . 2 | Σ j ℓ +1 | , we take G ∈ Σ j ℓ +1 ( δ ) and consider a tuple of v ertices ( y , u, v ) such that • y / ∈ V ′ j , u ∈ V ′ j , v / ∈ V ′ j , • and { x j ℓ +1 , y } , { u, v } ∈ E ( G ), { x j ℓ +1 , u } , { y , v } / ∈ E ( G ). If we switch { x j ℓ +1 , y } , { u, v } 7→ { x j ℓ +1 , u } , { y , v } , (26) w e get a graph from Σ j ℓ +1 ( δ + 1). It giv es | Σ j ℓ +1 ( δ ) | · ( | N X ( x j ℓ +1 ) | − ( g j ( x j ℓ +1 ) + δ )) · ( h j out ± O (( d/k ) 0 . 3 k t )) − |X ∩ Σ j ℓ +1 ( δ ) | O ( d 2 n ) = | Σ j ℓ +1 ( δ + 1) | · ( g j ( x j ℓ +1 ) + δ + 1) · ( h j in ± O (( d/k ) 0 . 3 k t + d 2 )) . In a similar wa y , we apply the switching op eration ( 26 ) to G ∈ Σ j ℓ +1 ( δ ) and get a graph from Σ j ℓ +1 ( δ − 1), where • y ∈ V ′ j , u, v / ∈ V ′ j , • and { x j ℓ +1 , y } , { u, v } ∈ E ( G ), { x j ℓ +1 , u } , { y , v } / ∈ E ( G ). W e get | Σ j ℓ +1 ( δ ) | · ( g j ( x j ℓ +1 ) + δ ) · ( h j in ± O (( d/k ) 0 . 3 k t + d 2 )) = | Σ j ℓ +1 ( δ − 1) | · ( | N X ( x j ℓ +1 ) | − ( g j ( x j ℓ +1 ) + δ − 1)) · ( h j out ± O (( d/k ) 0 . 3 k t )) − |X ∩ Σ j ℓ +1 ( δ − 1) | O ( d 2 n ) . If ( | N X ( x j ℓ +1 ) | − g j ( x j ℓ +1 )) h j out > g j ( x j ℓ +1 ) h j in , then | Σ j ℓ +1 ( δ +1) | | Σ j ℓ +1 ( δ ) | > 1 − O ( k 1 . 7 /d 0 . 7 ), implying | Σ j ℓ +1 | / | Σ j ℓ | ≥ (1 − O ( k 1 . 7 /d 0 . 7 )) ( d/k ) 0 . 3 ( d/k ) 0 . 3 > ( d/k ) 0 . 2 . (27) Otherwise, | Σ j ℓ +1 ( δ − 1) | | Σ j ℓ +1 ( δ ) | > 1 − O ( k 1 . 7 /d 0 . 7 ), implying ( 27 ), as w ell. The pro of of Theorem 32 is now complete. 7 Pro of of Theorem 16 Let us ﬁx U ⊆ [ n ] of size u = | U | ≤ cn d . (W e may assume that d ≤ cn as the statemen t is v acuously true otherwise.) Let c ≥ c ′ := ud/n . F or i ≥ 1, denote ε i := 2 ln d 2 i − 1 · d + 50 c ′ ( d − 1) i − 1 . W e start from the following claim. Claim 33. F or every p ositive inte ger r such that ud ( d − 1) r − 1 ≤ cn , P  | S r ( U ) | ≥ (1 − 3 ln d/d − 100 c ) ud ( d − 1) r − 1  ≥ 1 − r X i =1  2 ec ′ ( d − 1) i − 1 ε i  ε i d ( d − 1) i − 1 u . 31 Pr o of. Fix a p ositiv e integer r such that d ( d − 1) r − 1 u ≤ cn . Assuming that all the edges in B r − 1 ( U ) \ E ( G n [ S r − 1 ( U )]) ha ve b een explored, let as explore the next la yer S r ( U ). F or i ≤ r , let S 1 i ( U ) b e the set of v ertices in S i ( U ) that ha ve degree 1 in B i ( U ) \ E ( G n [ S i ( U )]). W e assume, b y induction, that | S 1 r − 1 ( U ) | ≥ d ( d − 1) r − 2 u 1 − 2 r − 1 X i =1 ε i ! whenev er r ≥ 2 (if r = 1, w e do not imp ose any assumption). Enumerate the vertices so that S 1 r − 1 ( U ) = { x 1 , x 2 , . . . } . W e order arbitrarily the edges that we ha v e to explore from x j ’s outside of B r − 1 ( U ). Assume that w e hav e exp osed all adjacencies for x 1 , . . . , x j − 1 , and the ﬁrst ℓ − 1 edges of x j . Let S ′ ⊆ S r ( U ) be the set of neigh b ours of { x 1 , . . . , x j } outside of B r − 1 ( U ) that ha ve been explored so far. Let us sho w that the ℓ -th edge { x j , v } touches S ′ ∪ S r − 1 ( U ) with probabilit y at most 2 c ′ ( d − 1) r − 1 , whic h can b e done using switc hings. Consider the family of d -regular graphs on the vertex set [ n ] that con tain the explored edges. W e partition this family into t wo subfamilies, Σ 0 and Σ 1 . In Σ 0 the ℓ -th edge { x j , v } do es not touch S ′ ∪ S r − 1 ( U ) whereas in Σ 1 it do es. F or every G ∈ Σ 1 (with the ℓ -th edge { x j , v } identiﬁed) and any pair of v ertices ( z , w ) such that • { z , w } ∈ E ( G ), { x j , z } , { v , w } / ∈ E ( G ), • z , w / ∈ S ′ ∪ S r − 1 ( U ), w e may switch { x j , v } , { z , w } 7→ { x j , z } , { v , w } to get a graph from Σ 0 (with the ℓ -th edge { x j , z } iden tiﬁed). T o compare the sizes of Σ 1 and Σ 0 , w e need to estimate the num b er of triples ( G, z , w ), G ∈ Σ 1 , and the num ber of triples ( G, v , w ), G ∈ Σ 0 . F or a giv en G ∈ Σ 1 , the n umber of pairs ( z , w ) is at least d ( n − | B r − 1 ( U ) | − | S ′ | ) − d ( | B r − 1 ( U ) | + | S ′ | ) − 2 d 2 = d ( n − 2 | B r − 1 ( U ) | − 2 | S ′ | − 2 d ) ≥ d  n − 2 u d d − 2 ( d − 1) r − 1 − 2 ud ( d − 1) r − 1 − 2 d  . On the other hand, for a giv en G ∈ Σ 0 , the n umber of pairs ( v , w ) is clearly at most ( | S r − 1 ( U ) | + | S ′ | ) d ≤ d ( ud ( d − 1) r − 2 + ud ( d − 1) r − 1 ) . Th us, we get | Σ 1 | ( n − 2 ud ( d − 1) r − 1 − 2 ud ( d − 1) r − 1 / ( d − 2) − 2 d ) ≤ | Σ 0 | ( ud ( d − 1) r − 2 + ud ( d − 1) r − 1 ) , implying the desired probabilit y b ound | Σ 1 | | Σ 1 | + | Σ 0 | ≤ | Σ 1 | | Σ 0 | ≤ ud ( d − 1) r − 2 + ud ( d − 1) r − 1 n − 2 ud ( d − 1) r − 1 − 2 ud ( d − 1) r − 1 / ( d − 2) − 2 d = ud ( d − 1) r − 1 (1 + 1 / ( d − 1)) n (1 − 2 d ( d − 1) r u/ ( n ( d − 2)) − 2 d/n ) ≤  1 + 1 d − 1 + 3 d ( d − 1) r − 1 u n  d ( d − 1) r − 1 u n (1 − 2 c ) < 2 c ′ ( d − 1) r − 1 , 32 as needed. Therefore, the probabilit y that the n um b er of edges explored in this wa y that “do not lead to a new v ertex” (i.e. do not fall into S 1 r ( U )) is at least ε r d ( d − 1) r − 1 | U | is dominated b y P  Bin  d ( d − 1) r − 1 u, 2 c ′ ( d − 1) r − 1  > ε r d ( d − 1) r − 1 u  ≤  d ( d − 1) r − 1 u ε r d ( d − 1) r − 1 u  (2 c ′ ( d − 1) r − 1 ) ε r d ( d − 1) r − 1 u ≤  2 ec ′ ( d − 1) r − 1 ε r  ε r d ( d − 1) r − 1 u . If the second endp oin t of an explored edge lands in S 1 r − 1 ( U ) as well, it immediately rev eals some edge that ha ve not b een explored y et. Therefore, the ab o v e even t implies that | S 1 r ( U ) | ≥ ( d − 1) d ( d − 1) r − 2 u 1 − 2 r − 1 X i =1 ε i !! − 2 ε r d ( d − 1) r − 1 u = d ( d − 1) r − 1 u 1 − 2 r X i =1 ε i ! ≥  1 − 4 ln d d − 100 c ′ ( d − 1) r − 1  ud ( d − 1) r − 1 . By induction, w e get that the latter even t holds with probabilit y at least 1 − r X i =1  2 ec ′ ( d − 1) i − 1 ε i  ε i d ( d − 1) i − 1 u , completing the proof of the claim. In order to complete the proof of the lemma, it suﬃces to apply the union b ound o v er all u ≤ cn d , all r such that ud ( d − 1) r − 1 < cn , and all U ⊆ [ n ] of size u : X u

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