On the resilience of Hamiltonicity and optimal packing of Hamilton cycles in random graphs

On the resilience of Hamiltonicit y and optimal pac king of Hamilton cycles in random graphs Sonn y Ben-Shimon ∗ Mic hael Kriv elevic h † Benn y Sudak ov ‡ Abstract Let k = ( k 1 , . . . , k n ) b e a seq uence of n integ ers. F or an increasing monotone g raph prop erty P w e sa y that a base graph G = ([ n ] , E ) i s k -r esilient with respect to P if f or ev ery subgraph H ⊆ G such that d H ( i ) ≤ k i for every 1 ≤ i ≤ n the gra ph G − H p ossesses P . This notion naturally extends the idea of the lo c al r esilienc e of graphs recently initiated b y Sudako v and V u. In t his paper w e study the k -resilience of a t ypical graph from G ( n, p ) with resp ect to th e Hamilto nicity p roperty where we let p range ov er all v alues for which the base graph is exp ected to b e Hamil tonian. In p articular, w e prov e that for ev ery ε > 0 and p ≥ ln n +ln ln n + ω (1) n if a graph is sampled from G ( n, p ) th en with high probability remo ving from ea ch vertex of “small” d egree all incident edges b ut t wo and from any oth er v ertex at most a ( 1 3 − ε )-fraction of the incident edges will result in a Hamiltonian graph. Considering this generalized approac h to the notion of resilience all ow s to establish sev eral corollaries whic h improv e on t he b est kn o wn b ounds of Hamilto nicity related questions. I t implies th at for every p ositiv e ε > 0 and large enough v alues of K , if p > K ln n n then with h igh probability the local resilience of G ( n, p ) with respect to being Hamiltonia n is at least (1 − ε ) np/ 3, impro vin g on the previous b ound for th is range of p . An other implication is a result on optimal p ac king of edge disjoin t Hamilton cycles in a random graph. W e p ro ve t hat if p ≤ 1 . 02 l n n n then with high probability a graph G sampled f rom G ( n, p ) conta ins ⌊ δ ( G ) 2 ⌋ edge disjoint Hamilton cycles, extending the previous range of p for which this w as known to hold. 1 In tro duction A Hamilton cycle in a graph is a simple cy cle that tra verses all vertices of the gr aph. The study whether a graph contains a Hamilton cycle, or the Hamiltonicity graph prop erty , b ecame one of the main themes o f graph theory from the very b eginning. Deciding whether a gr aph is Hamiltonian w as one of the first problems that were proved to b e NP-Complete , giving some insight on the elusiveness of this problem. Although the computational problem of deter mining in “r easonable” time whether a g raph is Hamiltonian se ems hop eless the r oad do es no t need to stop there, as ther e are man y other na tural and interesting ques tions ab out this ∗ Sc hool of Computer Science, Raymond and Bev erly Sackler F acult y of Exact Sciences, T el Aviv Uni v ersity , T el Aviv 69978, Israel. E-mail: sonn y .b enshimon@cs.tau.ac.il. Research partially supp orted b y a F ara jun F oundation F ello wship. † Sc hool of Mathematical Sciences, Raymond and Bev erly Sac kl er F acult y of Exact Sciences, T el Avi v Univ ersity , T el Aviv 69978, Israel. E-mail: kriv elev@tau.ac.il. Research supp orted in part by a USA-Israel BSF grant and by a grant from the Israel Science F oundation. ‡ Departmen t of Mathematics, UCLA, Los Angles 90005, CA, USA. Email: bsudako v@math.ucla.edu. Researc h suppor ted in part b y NSF CAREER aw ard DMS-0812005 and b y USA-Israeli BSF gran t. 1 prop erty . F or one, given a pre scrib ed num ber of edges m , are most g raphs o n n vertices with m edg es Hamiltonian? Hence, the study of Hamiltonicity of r andom gr aphs seems like a natural approa c h to pursue along the pa th to understand this intricate prop er t y . Fixing m and selecting uniformly at ra ndom a g raph on n vertices with m edg es was indeed the origina l random graph mo del in tro duced b y Erd˝ os and R ´ enyi, but the most widely studied random gr aph mo del is the binomial random gr aph, G ( n, p ). In this mo del we start with n vertices, lab eled, say , b y 1 , . . . , n , a nd select a gr aph o n these n vertices b y going over all  n 2  pairs o f vertices, deciding independently with pr obability p for a pair to b e an edge. The pro duct probabilit y space nature gives this model a greater app eal than the original one , but they ar e indeed very muc h related, a nd in a sense, equiv alent (see mono graphs [7] and [15] for a thorough in tro ductio n to the sub ject of random gra phs). W e note that we will so metimes abuse the notation and use G ( n, p ) to denote b oth the distribution on the graphs just describ ed and a r andom sample from this distribution; which o f the tw o should b e clea r fro m the context. W e denote by H AM the gra ph prope rty of having a Hamilton c ycle. One of the corner stone results in the theor y of ra ndom gra phs is that of Bollob´ as [6] and of Ko ml´ os and Szemer´ edi [17] who proved that if G ∼ G ( n, p ) for p ≥ ln n +ln ln n + ω (1) n (where ω (1) is an y function tending to infinity with the n umber of vertices, n ) then with high pr obability (or w.h.p. for brevity) 1 G ∈ H AM . It is fairly easy to show (see e.g. [7, Chapter 3 ]) that if p ≤ ln n +ln ln n − ω (1) n then w.h.p. G contains at least one vertex of degree smaller than 2 (and hence is not Hamiltonian). Not only do es this g ive a very precise range of p for which the typical graph from G ( n, p ) is Hamiltonian, this idea sugges ts that the b ottleneck for H AM (a t least for small v alue s of p ) stems fr om the vertices of low deg ree. 1.1 T yp es of r esilience Let P b e a mono tone increa sing graph prop erty (i.e. a family of graphs on the sa me vertex set whic h is closed under the a ddition of edg es and isomor phism). Counting the minimal n umber of edg es one needs to remov e from a base graph G in order to obtain a gr aph not in P may , ar guably , seem like o ne of the most natural questions to consider. Indeed, this notion, which is now commonly denoted b y the glob al r esilienc e of G with resp ect to P (or the e dit di stanc e of G with re spect to P ) is one of the fundamental q uestions in extremal combinatorics. This field can b e traced back to the celebrated theorem of T ur´ an [23] which states (in this terminology) that the complete graph K n on n v ertices has global res ilience n 2 ( n r − 1) (assuming r divides n ) w ith r esp ect to co n taining a co p y of K r +1 . F or some gr aph prop erties of glo bal nature, s uc h as HAM or being connected, the r emov a l of all edges incident to a vertex of minimum degree is enough to destroy them, hence supplying a trivia l upper b ound on the globa l resilience of any graph with re spec t to these prop erties. F or such prop erties the notion of glo bal resilience do es not seem to co n vey wha t one would exp ect fr om such a distance measure. O ne would like to gain s ome c ont rol on the amoun t of edges inciden t to a single v ertex that can b e remov ed. T o pursue this approach, for a g iven base graph G = ([ n ] , E ) one w ould lik e to gain better understanding of the po ssible degree sequences of subgraphs H o f G for which the gra ph G − H p ossesses the prop erty P . Let a = ( a 1 , . . . , a n ) and b = ( b 1 , . . . , b n ) b e t wo sequences of n n um ber s. W e write a ≤ b if a i ≤ b i for every 1 ≤ i ≤ n . Given a graph G on vertex set [ n ], we denote its degree sequence b y d G = ( d G (1) , . . . , d G ( n )). 1 In this pap er, we say that a sequenc e of ev ents A n in a r andom graph mo del o ccurs w.h.p. if the probabilit y of A n tends to 1 as the n um ber of vertices n tends to infinity . 2 Definition 1. 1 . Le t G = ([ n ] , E ) b e a graph. Given a sequence k = ( k 1 , . . . , k n ) a nd a monotone incre asing graph prop erty P , we say that G is k -r esilient with resp ect to the pr ope rty P if for every subgr aph H ⊆ G such that d H ≤ k , we hav e G − H ∈ P . It was Sudako v a nd V u [22] who initiated the systematic study of such a notio n, albeit stated a little differently . In their or iginal work the ob ject of study w as the minimu m v a lue o f the maximum degree of a non k -resilient s equence. They coined this par ameter as the lo c al re silienc e of a graph with respect to P . W e will use the following notation to denote this parameter r ℓ ( G, P ) = min { r : ∃ H ⊆ G such that ∆( H ) = r and G \ H / ∈ P } . So, fo r lo cal r esilience, there is a uniform constra in t o n the num b er o f deletions of edge s incident to a single vertex. Although not explicitly , the study of lo cal resilience lays in the hea rt o f pr evious results in classica l graph theory . In fact, one of the cornerstone results in the s tudy of H AM is the classical theore m of Dirac (see, e .g., [1 2, Theo rem 10.1.1]) which states (in this terminology ) that K n has lo cal resilience n/ 2 with resp ect to H AM . As has already been po in ted out in [22] there seems to be a dualit y b etw een the global or lo cal natur e of the gra ph property at hand and the t ype of resilience that is mo re natural to co nsider. More sp ecifically , g lobal resilience seems to b e a more appr opriate no tion for s tudying lo c al prop erties (e.g. containing a copy of K k ), whereas for globa l p rop erties (e.g. b eing Hamiltonian), the study of lo cal resilience app ears to be more natural. This study of resilience has gained p opularity , and in a relatively shor t perio d of time quite a few research pap ers studied this and r elated distance notions [2 2, 13, 11, 18, 3, 5, 2 , 9, 19, 14, 4]. This evolving b o dy of resear c h explor ed the resilience with respect to man y graph prop erties , where the base graph of fo cus was mainly the binomial r andom graph G ( n, p ) and graphs fro m families o f pseudo-ra ndom graphs. In one of the subsequent pap ers of D ellamonica et. al. [11], a mor e refined version of lo cal r esilience was co nsidered. Consider a graph G with deg ree sequence d . The authors of [11] defined the lo cal res ilience of G with resp ect to P as the maximal v a lue of 0 ≤ α ≤ 1 for which a graph G ( α d )-r esilient with resp ect to P . This no tion is a little mo re robust than the or iginal definition of lo cal r esilience a s it can a lso deal with degree se quences of irregula r gr aphs. The term k -r esilience is a further ge neralization on the sa me theme. The main motiv atio n for studying this more genera l notion is tha t it allows to give a unified appro ach to several problems a s will be e xpo sed. 1.2 Previous w ork One of the first lo ca l resilience pro blems Sudako v and V u [22] conside red w as the loc al resilience pr oblem of G ( n, p ) with resp ect to HAM . They provided b oth an upp er and low er b ound for this parameter for almos t all the range of p . First, note t hat b y the thr eshold probabilit y fo r minim um degree 2, the local resilience parameter b ecomes interesting only for p ≥ ln n +ln ln n + ω (1) n (as for low er v alues o f p , it is by definition equa l to zero for non- Hamiltonian gr aphs, which w.h.p. is the case in this range). As an upp er b ound they prov ed that for every 0 ≤ p ≤ 1 w.h.p. r ℓ ( G ( n, p ) , H AM ) ≤ np 2 (1 + o (1)) (1) Maybe more imp ortantly , the low e r b ound they prov ed states that for every δ, ε > 0, if p ≥ ln 2+ δ n n then w.h.p. r ℓ ( G ( n, p ) , H AM ) ≥ np 2 (1 − ε ) whic h essen tially settles the problem for this range of p . Note tha t in fact this can b e viewed as a far reaching gener alization of Dirac’s theo rem. F r ieze a nd Kr ivelevic h in [13] 3 studied this problem for the range of p “ shortly after” G ( n, p ) b ecomes Hamiltonian w.h.p., but the lower bo und they o btained in this r ange is weaker. They prov ed that there exist a bsolute constants α, C > 0 such that for ev ery p ≥ C ln n n w.h.p. r ℓ ( G ( n, p ) , H AM ) ≥ αnp . Recen tly , the authors in [5] impr ov ed on the ab ov e by showing that fo r every ε > 0 there exists a n absolute constan t C > 0 suc h that if p ≥ C ln n n then w.h.p. r ℓ ( G ( n, p ) , H AM ) ≥ np 6 (1 − ε ). It is plaus ible that w.h.p. r ℓ ( G ( n, p ) , H AM ) = np 2 (1 ± o (1)) a s s o on as p ≫ log n n , but the above mentioned results still lea v e a gap to fill. In this work we make some progress on this fr ont , but, alas, we ar e unable to close the gap completely . A rela ted questio n is the n umber of edge- disjoint Hamilton cycles one can hav e in a graph. Nash-Willia ms [20] a sserted tha t Dirac ’s s ufficien t co ndition for a Hamilton cyc le in fact guarantees at least ⌊ 5 n 224 ⌋ edge- disjoint Hamilton cycles. Quite recently , Chr istofides et. al. [10], answ ering one of Na sh-Williams’ o riginal conjectures asymptotically , prov ed that minim um degree  1 2 + o (1)  n is sufficien t for the existence of  n 8  edge-disjoint Ha milton cycles in a graph. When cons idering random graphs, the current knowledge about packing of edge-disjoin t Hamilton cycle is even mor e satisfactory . Bo llob´ as and F rieze [8] showed that for every fixed r , if p ≥ ln n +(2 r − 1) ln ln n + ω (1) n , the minimal p for whic h δ ( G ( n, p )) ≥ 2 r , one can t y pically find r edge-disjoint Hamilton cycle s in G ( n, p ). Kim and W o rmald [16] established a similar result for ra ndom d -regular g raphs (for fixe d d ), proving that such gr aphs typically c ont ain ⌊ d 2 ⌋ edge-disjo in t Hamilton cycles. The previo us sta temen ts are of course b est p os sible, but invite the natural questio n of what happ ens when the minim um degr ee is allow ed to grow with n . Denote b y H δ the prop erty of a graph G to con tain ⌊ δ ( G ) / 2 ⌋ - edge disjoin t H amilton cycles. F rieze and Kr ivelevic h in [1 3], e xtending [8], sho w ed that if p ≤ (1+ o (1)) ln n n then w.h.p. G ( n, p ) ∈ H δ . They even conjectured that this pro per t y is in fact typical fo r the whole r ange of p . Conjecture 1.2 (F rieze and Krivelevich [13]) . F or every 0 ≤ p ( n ) ≤ 1 , w.h. p. G ( n, p ) has the H δ pr op erty. In this pap er, we are able to extend the range of p for whic h Co njecture 1.2 holds, but cannot resolve the conjecture completely . 1.3 Our Results As pr eviously men tioned, in this work w e explore the notion of k -r esilience of random g raphs w ith res pect to HAM . T o sta te our main result we need the following notation. Let G = ([ n ] , E ) b e a gra ph. F or every po sitive t we denote by D t = D t ( G ) = { v ∈ [ n ] : d G ( v ) < t } the subset of vertices of degree less than t . Denote by d = ( d 1 , . . . , d n ) the deg ree sequence o f G . F or every co nstant ε > 0 and t > 0 we define the (not necessarily int egral) sequence e d ( t, ε ) = ( e d 1 , . . . , e d n ) as follows: 1. e d v = d v − 2 for every v ∈ D t ( G ); 2. e d v = d v  1 3 − ε  for every v ∈ V 1 = V \ D t ( G ). Theorem 1. F or every ε > 0 and p ≥ ln n +ln ln n + ω (1) n w.h.p. G = ([ n ] , E ) ∼ G ( n, p ) wi th de gr e e se quenc e d is e d ( np 100 , ε ) -r esilient with r esp e ct to HAM . Note that Theor em 1 essen tially cov ers the whole range of relev ant v alues of p . Moreov er, if p is such that there exists some ε > 0 fo r whic h w.h.p. δ ( G ( n, p )) − 2 ≤ np 100  1 3 − ε  , then the result gives an exa ct lo cal resilience for this ra nge. On the o ne hand, every graph G sa tisfies r ℓ ( G, HAM ) ≤ δ ( G ) − 1 as in order 4 to leave the graph Hamiltonian after the deletion of edges , all de grees in the resulting graph must b e a t least 2. O n the other hand, Theorem 1 guarantees that in this range of p if G ∼ G ( n, p ) then w.h.p. removing any subgraph H ⊆ G of maximum de gree ∆( H ) ≤ δ ( G ) − 2 leav es a Ha miltonian graph. Theorem 2. If ln n +ln ln n + ω (1) n ≤ p ≤ 1 . 02 ln n n and G ∼ G ( n, p ) then w.h.p. r ℓ ( G, HAM ) = δ ( G ) − 1 . Pr o of. As was previously men tioned, in lig h t of Theor em 1 it is enough to pr ov e that for the giv en range of p w.h.p. δ ( G ( n, p )) − 2 ≤ np 100  1 3 − ε  for some ε > 0. W e use a basic r esult in the theory of r andom graphs due to Bo llob´ as (see e.g. [7, Chapter 3]) which asserts that if  n − 1 k  p k (1 − p ) n − 1 − k = ω  1 n  , then w.h.p. δ ( G ( n, p )) ≤ k . Note that if k is s uc h tha t w.h.p. δ ( G ( n, p )) ≤ k , then this is also true for every p ′ ≤ p due to monotonicity . Setting k = ln n 300 it suffices to pro ve that for p = 1 . 02 ln n n − k , w.h.p. δ ( G ( n, p )) ≤ k ≤ np 100  1 3 − 1 160  + 2, which follows fro m  n − 1 k  p k (1 − p ) n − 1 − k ≥ (1 − o (1))  e ( n − k ) p k  k · e − ( n − k ) p ≥ (1 − o (1)) n 1+ln 306 300 · n − 1 . 02 = ω  1 n  . This completes the pro of o f the theorem. Comparing Theorem 2 with the upp er b ound f rom (1) we see that the lo c al resilie nce drops from b eing equal to one less than the minimal degree in the beg inning of the range to being e qual to less than roughly half of it a s p grows. This is due to the fact that when p is small enough the appear ances of vertices whose deg ree is muc h smaller than the av erage degr ee crea te a b ottleneck for HAM (and many o ther gr aph prop erties). Not all is lost when p b eco mes larg er. Our main result implies that taking p to b e large enough such that w.h.p. there a re no v ertices of degree less than np 100 , the re mov al o f almo st one third of the incident edges at ev ery v ertex from a t ypica l random graph will le av e a graph whic h is Hamiltonian. Again, using straightforward calcula tions, which we omit, the following result is readily established. Theorem 3. F or every ε > 0 ther e ex ist s a c onstant C = C ( ε ) > 0 such that if p ≥ C ln n n then w. h.p. r ℓ ( G ( n, p ) , H AM ) ≥ np 3 (1 − ε ) . Note that The orem 3 in fact improv es on the b est known results of [5] for the loc al resilience of G ( n, p ) with resp ect to H AM in this r ange of p . Lastly we show ho w from Theor em 1 one can obtain the existence o f an optima l packing of Hamilton cycles in a typical r andom graph. Theorem 4. F or every p ≤ 1 . 02 ln n n w.h.p. G ( n, p ) has the pr op erty H δ . Pr o of. First note that in light of the result of F rieze and K rivelevic h [13], w e may assume that p ≥ (1 + o (1)) ln n n . W e claim that in the rang e (1 + o (1)) ln n n ≤ p ≤ 1 . 02 ln n n w.h.p. one can sequentially re mo ve a Hamilton cycle from a t ypical graph G ∼ G ( n, p ) for ⌊ δ ( G ) / 2 ⌋ − 1 ro unds lea ving the graph Hamiltonian. Indeed, ass ume that the assertion of Theorem 2 holds, then the remov a l of 0 ≤ i ≤ ⌊ δ ( G ) / 2 ⌋ − 1 edge-disjoint 5 Hamilton cycles from G is a r emov a l o f a 2 i - regular s ubgraph from G , and therefor e Theorem 2 asserts that the resulting graph must be Hamiltonia n. The remov al o f the last Hamilton cycle concludes the pro of. Although the impro vemen t of Theorem 4 r elative to the pre vious b est k nown b ound on p o f F r ieze and Krivelevich [13], may s eem quite insignificant, it should be stres sed tha t the method used in [13] cannot be made to work for p = (1 + ε ) ln n n for any fixed ε > 0, so the improv e men t presen ted here is more of a qualitative na ture. Alas , the methods pr esent ed here too cannot b e extended mu ch further, a s the degree sequence of the random g raph b ecomes more bala nced causing D np 100 to be the e mpt y s et. So, Conjecture 1.2 remains op en. 1.4 Organization The rest of the pape r is organized as follows. W e start with Section 2 where we state all the needed preliminaries that ar e used thro ughout the pro ofs of our results. Section 3 is devoted to showing why a graph with pseudo random prop erties is in fact k -r esilient to H AM , and Section 4 is dedicated to prove that a ll of the random- like prop erties nee ded in the previo us section appear w.h.p. in G ( n, p ). Section 5 is devoted to the pro o f of the main result of this pap er, na mely , The orem 1, and we conclude the pap er with some final remarks a nd open questions in Section 6 2 Preliminaries In this section w e provide t he necessar y background information needed in the cours e o f the proofs of the main results of this pa per . 2.1 Notation Given a graph G = ( V , E ), the neighb orho o d N G ( U ) of a subset U ⊆ V of v ertices is the set of v ertices defined by N G ( U ) = { v / ∈ U : ∃ u ∈ U. { v , u } ∈ E } , and the degree of a vertex v is d G ( v ) = | N G ( { v } ) | . W e denote by E G ( U ) the set of edges of G that hav e b oth endp oints in U , and by e G ( U ) its cardinality . Similarly , for tw o disjoint subsets of vertices U and W , E G ( U, W ) denotes the set of edges with an endpo in t in U and the other in W , and e G ( U, W ) its cardinality . W e will s ometime refer to e G ( { u } , W ) by d G ( u, W ). W e use the usual no tation of ∆( G ) and δ ( G ) to denote the resp ective maximum and minimum degree s in G . W e say that H is a sp ann ing sub gr aph of G (or s imply a subg raph, as a ll the subgraphs w e consider will b e spanning), and write H ⊆ G if the gr aph H = ( V , F ) has the sa me vertex s et as G and its edge set satisfies F ⊆ E . W e will denote b y ℓ ( G ) the leng th of a lo ngest pa th in G . Let R < n b e p ositive in teger s and f : [ R ] → R + . W e s ay that a g raph G = ( V , E ) on n vertices is a ( R, f )- exp ander if every U ⊆ V o f car dinality | U | ≤ R s atisfies | N G ( U ) | ≥ f ( | U | ) · | U | . When f is a constant function equal to some β > 0 w e say that G is a ( R, β )-expander . When a function f : A → R + satisfies f ( a ) ≥ c for any a ∈ A , where c ≥ 0 is a constant, we simply write f ≥ c . R emark 2.1 . Note that if G = ( V , E ) is a n ( R, f )-expander, then ev ery H = ( V , F ) fo r F ⊇ E is a lso an ( R, f )-expander. 6 The main research int erest o f this paper is the asymptotic behavior of some proper ties of graphs, when the gra ph is sa mpled from some probability meas ure G o ver a set of gr aphs on the same vertex set [ n ], and the num ber of v e rtices, n , grows to infinit y . Therefo re, from now o n a nd thro ughout the rest of this work, when nee ded we will alw ays as sume n to be large eno ugh. W e use the usual as ymptotic notation. F or tw o functions of n , f ( n ) and g ( n ), we denote f = O ( g ) if there exists a constant C > 0 such that f ( n ) ≤ C · g ( n ) for larg e enough v alues o f n ; f = o ( g ) o r f ≪ g if f / g → 0 as n go es to infinity; f = Ω( g ) if g = O ( f ); f = Θ( g ) if b oth f = O ( g ) and g = O ( f ). Throughout the paper w e will need to employ b ounds on large deviations of random v ar iables. W e will mostly use the following well-kno wn bo und on the low er and the upp er tails o f the binomial distr ibution due to Cher noff (see e .g. [1, App endix A]). Theorem 2.2 (Chernoff bo unds) . L et X ∼ Bin ( n, p ) , then for every α > 0 1. Pr [ X > (1 + α ) np ] < exp( − np ((1 + α ) ln(1 + α ) − α )) ; 2. Pr [ X < (1 − α ) np ] < exp( − α 2 np 2 ) ; It will sometimes be more co n venien t to use the following b ound o n the uppe r tail of the binomial distribution. Lemma 2.3. If X ∼ Bin ( n, p ) a nd k ≫ np , then Pr [ X ≥ k ] ≤  n k  p k ≤ ( enp/k ) k . Lastly , we stres s that thr oughout this pap er w e may omit floo r and ceiling v alues when these are not crucial to avoid cum b ersome exp osition. 3 F rom pseudorandomness to Hamiltonicit y This section will pr ovide all the necessary steps to show why a gr aph whic h po ssesses some r andom-like prop erties must b e Hamiltonia n. Firs t, we show that a pseudora ndom graph must contain an expander subgraph (Lemma 3.2). W e will a lso insist this expander subgraph is quite spars e in relatio n to the orig inal graph, where the sparsity requirement on the expander subgr aph will play a ma j or role in the pro of of Theorem 1. As a second step we show ho w the expansion of small sets of vertices is a combinatorial pro per t y which is quite reso urceful tow a rds proving Hamiltonicity , where the main to ol used to achiev e this is the celebrated P´ os a’s rota tion-extension technique (Lemma 3.6). 3.1 F rom pseudorandomness to expansion W e star t by defining a family of graphs with desired pseudorandom prop erties. Definition 3.1. W e sa y that a gra ph G 1 = ( V 1 , E 1 ) on n 1 vertices is ( n 1 , d, β )- quasi-r andom if it s atisfies the following prop erties: (P0) δ ( G 1 ) ≥ d 150 ; (P1) Every U ⊆ V 1 of cardinality | U | < n 0 . 11 1 ln n 1 satisfies e G 1 ( U ) ≤ d 0 . 13 | U | ; 7 (P2) Every U ⊆ V 1 of cardinality | U | < 12 β n 1 satisfies e G 1 ( U ) ≤ 5 0 β d | U | ; (P3) Every t wo disjoint subsets U, Z ⊆ V 1 where | U | = β n 1 and | Z | =  1 3 − 27 β  n 1 satisfy e G 1 ( U, Z ) ≥ n 1 ln ln n 1 . The goal will b e to find subgr aphs of ( n 1 , d, β )-quas i-random graphs which sa tisfy certain expansion prop erties. T o define this family of subgraphs we intro duce the follo wing “expansion” function. Given an int eger n 1 , we let f β : [ β n 1 ] → R + denote the function defined by: (Q1) f β ( t ) = (ln n 1 ) 0 . 8 for every integer 1 ≤ t < n 0 . 1 1 ; (Q2) f β ( t ) = 11 for every integer n 0 . 1 1 ≤ t < β n 1 ; (Q3) f β ( β n 1 ) = 2(1+39 β ) 3 β . The follo wing lemma gua rantees that ev ery ( n 1 , d, β )-quas i-random graph co n tains a sparse ( β n 1 , f β )-expander subgraph. Lemma 3. 2. F or every c onstant 0 < β < 1 15 · 10 4 ther e exists an inte ger n 0 = n 0 ( β ) > 0 such t hat if G 1 is an ( n 1 , d, β ) -quasi-r andom gr aph for some n 1 ≥ n 0 and d ≥ ln n 1 then G 1 c ontains a ( β n 1 , f β ) -exp ander sub gr aph Γ satisfy ing e (Γ) ≤ 10 6 β e ( G 1 ) . Pr o of. Pick every edge o f G 1 = ( V 1 , E 1 ) to be an e dge o f Γ with pro bability γ = 15 · 1 0 4 β indepe nden tly of all other choices. O ur g oal is to prove that Γ is an ( β n 1 , f β )-expander with po sitive probability . First, we a nalyze the minimum degr ee of Γ. The degree of ev ery vertex v ∈ V 1 in Γ is binomially distributed, d Γ ( v ) ∼ Bin ( d G 1 ( v ) , γ ), with median at least ⌊ γ δ ( G 1 ) ⌋ . Therefore Pr [ d Γ ( v ) ≥ ⌊ γ δ ( G 1 ) ⌋ ] ≥ 1 / 2. W e choos e n 1 to be lar ge enough so tha t ⌊ γ δ ( G 1 ) ⌋ ≥ γ d 200 , then b y pro per t y P0 a nd the fact that the degr ees in Γ of every tw o vertices are p os itiv ely correlated, w e have that Pr  δ (Γ) ≥ γ d 200  ≥ Pr [ ∀ v ∈ V 1 . d Γ ( v ) ≥ ⌊ γ δ ( G 1 ) ⌋ ] ≥ Y v ∈ V 1 Pr [ d Γ ( v ) ≥ ⌊ γ δ ( G 1 ) ⌋ ] ≥ 2 − n 1 , using the FKG inequality (see e.g. [1, Cha pter 6]). Under the a ssumption that δ (Γ) ≥ γ d 200 we show that Γ must sa tisfy Q1 , namely that every U ⊆ V 1 of cardinality | U | < n 0 . 1 1 satisfies | N Γ ( U ) | ≥ | U | (ln n 1 ) 0 . 8 . Let U ⊆ V 1 be some subset of cardinality | U | < n 0 . 1 1 , and a ssume that | N Γ ( U ) | < | U | (ln n 1 ) 0 . 8 . Denote by W = U ∪ N Γ ( U ), then b y our ass umption | W | < | U | (1 + (ln n 1 ) 0 . 8 ) ≤ | U | (ln n 1 ) 0 . 81 ≪ n 0 . 11 1 ln n 1 . Pro per t y P1 of G 1 implies b oth e Γ ( U ) ≤ e G 1 ( U ) ≤ d 0 . 13 | U | and e Γ ( W ) ≤ e G 1 ( W ) ≤ d 0 . 13 | W | < | U | d 0 . 13 (ln n 1 ) 0 . 81 . On the other hand, e Γ ( W ) ≥ δ (Γ) · | U | − e Γ ( U ) ≥ γ d 200 · | U | − e G 1 ( U ) > | U | d 0 . 13 ( γ (ln n 1 ) 0 . 87 200 − 1) > | U | d 0 . 13 (ln n 1 ) 0 . 81 which is a c ont radiction. Prop erty Q2 of Γ will follow the exact same lines, again under the a ssumption that δ (Γ) ≥ γ d 200 . Let U ⊆ V 1 be so me subset of car dinalit y n 0 . 1 1 ≤ | U | < β n 1 , a nd as sume that | N Γ ( U ) | < 11 | U | . Denote by W = U ∪ N Γ ( U ), then by our assumption | W | < 1 2 | U | < 12 β n 1 . Pro per t y P2 of G 1 implies b oth e Γ ( U ) ≤ e G 1 ( U ) ≤ 50 β d | U | and e Γ ( W ) ≤ e G 1 ( W ) ≤ 50 β d | W | < 600 β d | U | . On the other hand, e Γ ( W ) ≥ δ (Γ) · | U | − e Γ ( U ) ≥ γ d 200 · | U | − e G 1 ( U ) ≥ 7 50 β d | U | − 50 β d | U | > 600 β d | U | which is a contradiction. W e pr o ceed to show that Γ satisfies the b ound on its num b er o f edges and prop erty Q3 with pr obability greater than 1 − 2 − n 1 . This will imply that there ex ists a subgr aph Γ as stated b y the lemma with po sitive 8 probability . Consider the num ber of edges in Γ. Clearly , e (Γ) ∼ Bin ( e ( G 1 ) , γ ) and by pr op erty P0 we hav e e ( G 1 ) ≥ n 1 δ ( G 1 ) 2 ≥ n 1 d 300 ≥ n 1 ln n 1 300 . Using L emma 2.3 it follows that Pr  e (Γ) > 1 0 6 β e ( G 1 )  ≤  e · e ( G 1 ) · γ 10 6 β e ( G 1 )  10 6 β e ( G 1 ) = o (2 − n 1 ) . Fix a subset U ⊆ V of cardina lit y | U | = β n 1 . Ass ume that for ev er y s ubset of vertices Z ⊆ V 1 of cardi- nality | Z | =  1 3 − 27 β  n 1 and disjoint from U , we have that e Γ ( U, Z ) > 0 , then | N Γ ( U ) | ≥  2 3 + 26 β  n 1 = 2(1+39 β ) 3 β | U | . Therefore, in or der to pr ov e that Γ satisfies Q 3 with the req uired proba bilit y , w e pr ov e that e Γ ( U, Z ) > 0 fo r every pair of disjoint subsets of vertices U, Z ⊆ V 1 of ca rdinality | U | = β n 1 , and | Z | =  1 3 − 27 β  n 1 with probability at least 1 − 2 − n 1 . By prop erty P3 it follows that e G 1 ( U, Z ) > n 1 ln ln n 1 , and ther efore e Γ ( U, Z ) is sto chastically dominated by Bin ( n 1 ln ln n 1 , γ ). This implies that Pr [ e Γ ( U, Z ) = 0] < (1 − γ ) n 1 ln ln n 1 < exp( − γ n 1 ln ln n 1 ) . Now, b y going ov er all po ssible pairs of subsets U and Z , as there a re at most 4 n 1 of those, we have that the probability that Γ do es not satisfy prop er t y Q3 is a t most exp( − γ n 1 ln ln n 1 ) · 4 n 1 = o (2 − n 1 ) which completes the pro of. 3.2 F rom expansion to Hamiltonicit y: P´ osa’s r o t ation-extension technique In or der to descr ibe the relev ant connection b etw een Hamiltonicity and expanders, we r equire the notion of b o osters . Definition 3.3. F or every graph G we say tha t a non-edge { u, v } / ∈ E ( G ) is a b o oster with resp ect to G if G + { u, v } is Hamiltonian or ℓ ( G + { u, v } ) > ℓ ( G ). F or any vertex v ∈ V we deno te b y B G ( v ) = { w / ∈ N G ( v ) ∪ { v } : { v , w } is a b o os ter } . (2) The follo wing simple lemma describ es a sufficien t condition for a graph G to be such that deleting the edges o f some subgraph H ⊆ G leav es a Hamiltonian gra ph. Note that a crucial p oint in the lemma is the existence of another graph w hic h a cts as a “ba ckbone” by pr oviding eno ugh b o oster s. Lemma 3.4. L et G = ( V , E ) b e a gr aph and let H ⊆ G b e some sub gr aph of it. If ther e ex ists a sub gr aph Γ ⊆ G − H su ch t hat for every E ′ ⊆ E ( G − H − Γ) of c ar dinality | E ′ | ≤ n ther e ex ist s a vertex v ∈ V (which may dep end on E ′ ) s atisfying | N G − (Γ+ E ′ ) ( v ) ∩ B Γ+ E ′ ( v ) | > d H ( v ) , then G − H is Hamiltonian. Pr o of. Assume Γ ⊆ G − H is as r equired by the lemma, then we prov e there exists an edge set F ⊆ E ( G − H − Γ) such that the gr aph Γ + F must b e Hamiltonian, which in tur n implies G − H is Hamiltonia n. Start with F 0 = ∅ . Assume that F i is a subset o f 0 ≤ i ≤ n e dges of E ( G − H − Γ). If the gr aph Γ i = Γ + F i ⊆ G − H is Hamiltonia n we are do ne. Otherwise, by the as sumption of the lemma, there exists a vertex v i ∈ V such that | N G − Γ i ( v i ) ∩ B Γ i ( v i ) | > d H ( v i ), and hence there ex ists at least o ne neighbor of v i in G , which w e denote b y w i , such that the pair { v i , w i } is still an edge in G − H , and is a b o oster with re spect to Γ i (hence no t an edge o f Γ i ). It follows that the graph Γ i + { v i , w i } is Hamiltonian or ℓ (Γ i + { v i , w i } ) > ℓ (Γ i ). Finally , set F i +1 = F i ∪ {{ v i , w i }} . Note that there must exis t an integer i 0 ≤ n such that Γ i 0 is Ha miltonian, as the length of a path cannot exceed n − 1 . 9 W e no w describe and apply a crucial technical to ol, originally developed by P ´ osa [21], whic h lies in the foundation of many Hamilto nicit y results of ra ndom and pse udo-random gr aphs. This technique, which has come to b e known as P´ osa’s ro tation-extension, relies on the following basic op eration on a longest path in a graph. W e use the following tw o definitions. Definition 3.5. Le t G = ( V , E ) b e a gra ph, and let P = ( v 0 , v 1 , . . . , v ℓ ) be a longest pa th in G . If { v i , v ℓ } ∈ E for some 0 ≤ i ≤ ℓ − 2, then a n elementary ro tation of P along { v i , v ℓ } is the construction of a new longest path P ′ = P − { v i , v i +1 } + { v i , v ℓ } = ( v 0 , v 1 , . . . , v i , v ℓ , v ℓ − 1 , . . . , v i +1 ). W e say that the edge { v i , v i +1 } is br oken b y this rotation. Given a fixed β > 0 and integer n , we define a family L = L ( n, β ) of g raphs with vertex set V = [ n ] to which these elementary rotations will b e a pplied to. The graphs in L a ll entail so me pseudo-r andom prop erties. Now, a gr aph G = ( V , E ) ∈ L ( n, β ) if its vertex set can b e partitioned V = V 1 ∪ D such that: (L1) | D | ≤ n 0 . 09 ; (L2) d ( u, V 1 ) ≥ 2 for every u ∈ D ; (L3) There is no path in G of length at most 2 ln n 3 ln ln n with bo th (p oss ibly identical) endpoints in D ; (L4) G 1 = G [ V 1 ] is a ( β n 1 , f β )-expander on n 1 = | V 1 | vertices. Using elemen tary rotations w e proceed to show t hat an y G ∈ L ( n, β ) must be Hamiltonian or that the subset of vertices with “large” B G ( v ) must also b e large. Our pro of uses similar ideas to those found in [5]. Lemma 3.6. F or every fixe d β > 0 ther e exists an inte ger n 0 ( β ) = n 0 > 0 su ch that if n ≥ n 0 then every gr aph G = ( V , E ) ∈ L ( n, β ) is H amiltonian or must satisfy |{ v ∈ V 1 : | B G ( v ) | ≥ n/ 3 + β n }| ≥ n/ 3 + β n . R emark 3.7 . W e remark that prop erty L4 implicitly implies an upper bound on β for L ( n, β ) to be non- empt y . Indeed 2(1+39 β ) 3 ≤ 1 − β m ust hold which implies β ≤ 1 81 . Pr o of. First, w e claim that under the as sumptions o n G , it must b e connected. In order to pro v e this, we beg in b y showing that G 1 is connected. Indeed, assume otherwise and let W ⊆ V 1 be a connected comp onent of car dinalit y | W | ≤ n 1 / 2. Prop er ties Q1 and Q3 imply that every subset of at mos t β n 1 vertices has a non-empty neighbo r se t, hence we can further assume that | W | > β n 1 . L et W ′ ⊆ W b e of car dinality β n 1 , then | N G ( W ′ ) | > 2 n 1 / 3 > n 1 / 2 by prop erty Q3 and hence cannot b e contained in W , a contradiction. Prop erty L2 guarant ees that every vertex in D is connected to some vertex in V 1 , so adding the vertices of D to the graph G 1 leav e s the graph connected. T ake a longest path P = ( v 0 , . . . , v ℓ ) in G . By the assumption on G we ca n cle arly a ssume that ℓ ≥ 3 . Since P is a long est path, N G ( v 0 ) ∪ N G ( v ℓ ) ⊆ P . W e fir st claim that we can choo se such a pa th P with b oth of it s endpoints in V 1 . So, assume that at least one of the endp oints of P , v 0 and v ℓ , is in D . If v 0 and v ℓ are neig h b ors then G must contain a cycle of length ℓ ( G ). This implies tha t G is Hamiltonia n, as otherwise ℓ ( G ) < n and since G is connected there is an edg e emitting out of this cycle cr eating a path of length ℓ ( G ) + 1 in G which is a contradiction. W e can thu s assume v ℓ and v 0 are not b e neighbors. Assume w.l.o.g. v ℓ ∈ D then by pro per t y L2 the vertex v ℓ m ust hav e a neighbor other than v ℓ − 1 . As all the neigh bo rs of v ℓ m ust lie on P we denote this neighbor by v i where 1 ≤ i ≤ ℓ − 2. Performing an elementary rotation of P along { v i , v ℓ } results in the path P ′ = ( v 0 , v 1 , . . . , v i , v ℓ , v ℓ − 1 . . . , v i +1 ) with v i +1 as an endp oint. B y 10 Prop erty L3 of D , the v ertex v i +1 / ∈ D since it is at distance at most 2 fro m v ℓ ∈ D . Denote the resulting longest pa th whic h terminates at v i +1 by P ′ . If v 0 ∈ D as well then we can also p erfor m a rotation on P ′ keeping v i +1 fixed. Note that in o rder to do so we can ass ume v 0 and v i +1 are non-neig h b ors, as otherwise our graph in Hamiltonian a s previously claimed. W e can th us assume that our initial lo ngest path P has indeed t wo endp oints in V 1 . W e se t with for esight r 0 = ln n 1 7 ln ln n 1 , a nd r to b e the minimal in teger for which β n 1 ≤ 2 r − r 0 · T r 0 < 2 β n 1 , where T = ln 0 . 7 n 1 . F o r ev ery 0 ≤ i ≤ r w e will find a set S i ⊆ V 1 which is compos ed of (not necessarily all) endp oint s of longest paths in G obtained by per forming a s eries of i ele men tary rotations starting from P while keeping the endpoint v 0 fixed such that for every j < i in the j -th rotation the non- v 0 endpo in t lies in S j ⊆ V 1 . W e construct this sequence of sets { S i } r i =0 as follows. W e start with S 0 = { v ℓ } (which, by our assumption, is not in D ). F or every i > 0, let U i = { v k ∈ N G ( S i ) : v k − 1 , v k , v k +1 / ∈ S i j =0 S j } . Let v k ∈ U i , a nd x ∈ S i be such that { x, v k } ∈ E , and denote by Q the longest path from v 0 to x obtained from P by i elemen tary rotations leaving v 0 fixed. By the definition of U i , no ne o f the vertices v k − 1 , v k , v k +1 is an endpo in t of one of the se quence of longest paths starting from P and yielding Q , hence b oth edges { v k − 1 , v k } and { v k , v k +1 } w ere not broken and ar e ther efore pr esent in Q . Rotating Q along the edg e { x, v k } will make one of the vertices { v k − 1 , v k +1 } an endp oint in the resulting path. Assume w.l.o .g. that it is v k − 1 , a nd hence add it to the set S ′ i +1 . Note that the vertex v k − 1 can also b e added to the set S ′ i +1 if the vertex v k − 2 in U i , therefore | S ′ i +1 | ≥ 1 2 | U i | ≥ 1 2   | N G ( S i ) | − 3 i X j =0 | S j |   . (3) Having defined the set S ′ i +1 for every i , w e pro ceed to demonstrate how the set S i +1 is chosen as an appropria te subset o f S ′ i +1 . This is do ne a ccording to the v alue of i . First, for 0 ≤ i < r 0 the s et S i is chosen to b e o f cardinality T i , where we prove w e can do so inductively . Clea rly , our assumption o n S 0 prov es that the ba se of the induction holds . Assuming we can choose the subsets S j ⊆ S ′ j for a ll j ≤ i < r 0 − 1, we prov e we can choo se a subset S i +1 ⊆ S ′ i +1 ∩ V 1 . W e hav e by our induction hypo thesis that | S i | = T i < T r 0 ≤ n 0 . 1 1 , hence for every i < r 0 prop erty Q1 implies | N G ( S i ) | ≥ T i +1 . 1 , and ther efore b y (3) | S ′ i +1 | ≥ 1 2  T i +1 . 1 − 3  T i +1 − 1 T − 1  > T i +1 + 1 . Every vertex in S ′ i +1 is at distance a t most 2 i fr om v ℓ , therefor e, every tw o vertices in S ′ i +1 are at distance at most 4 i < 4 r 0 ≤ 4 ln n 1 7 ln ln n < 2 ln n 3 ln ln n from each other. P rop erty L2 implies there ca n be a t most one vertex from D in this set. Removing this v ertex if it exists, w e can set S i +1 to b e any subset of S ′ i +1 of T i +1 vertices from V 1 . In particular | S r 0 | = T r 0 ≫ n 0 . 09 ≥ | D | . Next, for r 0 ≤ i < r we will cons truct the sets S i to b e of cardinality R · 2 i − r 0 ≫ | D | , where R = T r 0 . Prop erties Q1 and Q2 imply | N G ( S i ) | ≥ 11 · | S i | ≥ 10 | S i | + 2 | D | , and ther efore b y (3) | S ′ i +1 | ≥ 1 2   10 · R · 2 i − r 0 + 2 | D | − 3   r 0 − 1 X j =0 T i + i X j = r 0 R · 2 j − r 0     ≥ | D | + 5 R 2 i − r 0 − 3 2  T r 0 + R (2 i +1 − r 0 − 1)  = | D | + 5 R 2 i − r 0 − 3 R 2 i − r 0 ≥ R · 2 i − r 0 + | D | . W e can therefor e set S i +1 to b e any s ubset of S ′ i +1 ∩ V 1 of ca rdinality R · 2 i − r 0 . F o r the sake o f conv enience we repla ce S r by some ar bitrary subset of it of cardinality β n 1 and denote this new set by S r . Fina lly , 11 we constr uct similarly the set S ′ r +1 . By prop erty Q3 we ha ve | N G ( S r ) | ≥ (1 + 39 β )2 n 1 / 3. Just as in the previous cases b y (3) | S ′ r +1 | ≥ 1 2   2 n 1 3 (1 + 3 9 β ) − 3   r 0 − 1 X j =0 T i + r X j = r 0 R · 2 j − r 0     ≥ n 1  1 3 + 13 β  − 3 2 T r 0 − 3 2 R · 2 r − r 0 +1 ≥ n 1  1 3 + 13 β  − o ( n 1 ) − 6 β n 1 > n 1 3 + 6 β n 1 > n 3 + β n. Assume S ′ r +1 ∩ N G ( v 0 ) 6 = ∅ , then G m ust co n tain a cycle of length ℓ ( G ). This implies that G is Hamilto- nian. Assume otherw ise0, then ℓ ( G ) < n and since G is connected there is a n edge emitting out of this cycle, creating a path of length ℓ ( G ) + 1 in G whic h is a con tradiction. This implies that S ′ r +1 ⊆ B G ( v 0 ). Now take an y endpo in t u 0 in S ′ r +1 and take a longest path P ′ starting from u 0 (whic h must exist since all vertices of S ′ r +1 are endpo in ts o f long est paths starting in v 0 ) and r epea t the same argument, while r otating P ′ and keeping u 0 fixed. This w ay we o btain the desired set | B G ( u 0 ) | o f n/ 3 + β n endpoints for every u 0 ∈ S ′ r +1 , th us completing the pro of. 4 Structural prop erties of G ( n, p ) W e star t with a very s imple claim r egarding the num b er of edges in the binomia l ra ndom g raph mo del G ( n, p ). Claim 4.1. F or every p ≥ ln n n w.h.p. n 2 p 4 ≤ e ( G ( n, p )) ≤ n 2 p . Pr o of. This is a simple application of Theorem 2.2. Clearly , e ( G ) ∼ Bin  n 2  , p  then Pr  e ( G ) > n 2 p  < Pr  e ( G ) ≥ 2  n 2  · p  ≤ exp( − ( n − 1) ln n 6 ) = o (1). Similarly , P r h e ( G ) < n 2 p 4 i ≤ Pr  e ( G ) < 0 . 51  n 2  p  ≤ exp  0 . 49 2 ( n − 1) ln n 4  = o (1). The following is (a special case of ) a well known pro per t y o f the binomial r andom gr aph mo del. It provides a very precise answ e r to which v alues of p do es a typical graph in G ( n, p ) ha s minimum degree at least 2 (see e .g. [7]). Theorem 4.2. L et h ( n ) = ω (1) b e any function which gr ows arbitr arily slow ly with n , then 1) if p ≥ ln n +ln ln n + h ( n ) n then w. h.p. δ ( G ( n, p )) ≥ 2 ; 2) if p ≤ ln n +ln ln n − h ( n ) n then w. h.p. δ ( G ( n, p )) < 2 . Recall that given a g raph G w e defined the set D t ( G ) = { v ∈ V : d G ( v ) < t } . W e prove some structura l prop erties of the se t D t ( G ) where G is sa mpled from the ra ndom gra ph mo del G ( n, p ). Claim 4.3. F or every p ≥ ln n n and inte ger t ≤ np 100 w.h.p. |D t ( G ( n, p )) | ≤ n 0 . 09 . 12 Pr o of. Let G ∼ G ( n, p ), then setting t 0 = np 100 , we can bound the pro bability that a vertex is in D t ( G ) as follows. Pr [ d G ( v ) < t ] ≤ Pr [ Bin ( n − 1 , p ) < t 0 ] ≤ t 0 − 1 X i =0  n − 1 i  p i (1 − p ) n − 1 − i ≤ t 0 ·  n − 1 t 0  p t 0 (1 − p ) n − 1 − t 0 ≤ t 0 ·  e ( n − 1 ) p t 0  t 0 e − p ( n − 1 − t 0 ) ≤ exp  − np + p + np 2 100 + np 100 (1 + ln 100) + ln np 100  < e − 0 . 92 np ≤ n − 0 . 92 . This implies tha t E [ D t ( G ( n, p ))] ≤ n 0 . 08 , and the cla im follows fro m Mar k ov’s inequality . Claim 4.4. F or every p ≥ ln n n and inte ger t ≤ np 100 , w.h.p G ∼ G ( n, p ) do es not c ontain a non-empty p ath of le ngth a t most 2 ln n 3 ln ln n such that b oth of its (p ossibly identic al) endp oints lie in D t ( G ) . Pr o of. Setting t 0 = np 100 , we prov e the claim for tw o distinct endpoints in D t 0 ( G ), and for paths o f length r where 2 ≤ r ≤ 2 ln n 3 ln l n n . The other cases (i.e. identical e ndpoints or r = 1) are similar and a little simpler. Fix t wo v ertices u, w ∈ V ( G ) and let P = ( u = v 0 , . . . , v r = w ) b e a se quence of vertices of V ( G ), where 2 ≤ r ≤ 2 ln n 3 ln l n n . Denote by A P the even t { v i , v i +1 } ∈ E ( G ) for ev er y 0 ≤ i ≤ r − 1, and by B u,w the event that bo th u and w are elements of D t 0 ( G ). Cle arly , Pr [ A P ] = p r , then Pr [ B u,w ∧ A P ] = p r · Pr [ B u,w | A P ] . Let X u,w denote the ra ndom v ariable which coun ts the n um ber of edges in G incident with u or w disrega rding the pa irs { u , v 1 } , { v r − 1 , w } , and { u , w } . W e can therefore bound Pr [ B u,w | A P ] ≤ Pr [ X u,w < 2 t 0 − 2] and as X u,w ∼ Bi n (2 n − 6 , p ), it follows that Pr [ X u,w < 2 t 0 − 2] ≤ 2 t 0 − 2 X i =0  2 n − 6 i  p i (1 − p ) 2 n − 6 − i ≤ np 50  2 n − 6 2 t 0 − 2  p 2 t 0 − 2 (1 − p ) 2 n − 4 − 2 t 0 ≤ np 50 ·  e (2 n − 6) p 2 t 0 − 2  2 t 0 e − p (2 n − 4 − 2 t 0 ) ≤ exp  − 2 np + p  4 + np 50  + np 50 (2 + ln 100) + ln np 50  < e − 1 . 8 np . Fixing the tw o e ndpoints u, w , the num ber of suc h s equences is a t mos t ( r − 1)!  n r − 1  ≤ n r − 1 . Applying a union bound argument over all suc h pa irs of vertices and p ossible sequences connecting them w e conclude that the proba bilit y there exists a path in G of length r ≤ 2 ln n 3 ln l n n , c onnecting t w o v ertices of D t ( G ) is at 13 most 2 ln n 3 ln ln n X r =1  n 2  · n r − 1 · p r · e − 1 . 8 np ≤ 2 ln n 3 ln ln n X r =1 n r +1 2 · ln r n n r · n − 1 . 8 ≤ ln n 3 ln ln n · n − 0 . 8 · (ln n ) 2 l n n 3 l n ln n = o (1) , where the first inequality follows by noting that the expression on l.h.s. decreases as p gro ws, hence we can replace p b y ln n n . This co mpletes the pro o f of the claim. Given a gr aph G and some t > 0 (which may dep end on n and p ), we denote by G 1 ( t ) = G 1 = ( V 1 , E 1 ) the gra ph G 1 = G [ V \ D t ( G )] and by n 1 its num b er o f v ertices, i.e. | V 1 | = | V \ D t ( G ) | = n 1 . The following lemma, which contains the main technical r esult of this section, implies that if p ≥ ln n n then the remov al of the vertices of low degree from a t y pical graph G sampled from G ( n, p ) leav es a gr aph G 1 which is robust in the following sense : The deletion of almost a third of the edges at each vertex of G 1 leav e s a g raph with some strong pse udo-random prop erties. Lemma 4.5. F or every fi x e d ε > 0 , ther e exists a smal l enough c onst ant β 0 = β 0 ( ε ) > 0 su ch t hat for ev ery 0 < β ≤ β 0 , p ≥ ln n n and t = np 100 , if G ∼ G ( n, p ) then w.h .p. for any sub gr aph H 1 ⊆ G 1 ( t ) = G 1 such that d H 1 ≤ ( 1 3 − ε ) d G , the gr aph G 1 − H 1 is ( n 1 , np, β ) -qu asi-r andom. Pr o of. First, by C laim 4.4 we can assume that in G ev e ry vertex of V 1 has at mo st one neighbor in D t ( G ), hence for every v ∈ V 1 we ha ve d G 1 ( v ) ≥ d G ( v ) − 1, and therefore δ ( G 1 ) ≥ t − 1. It follows that δ ( G 1 − H 1 ) >  2 3 + ε  δ ( G 1 ) > 2 t 3 = np 150 . Sec ond, using Claim 4.3 w e can, and will, assume that n 1 ≥ n − n 0 . 09 = n (1 − o (1)). The rest of the proper ties, and hence the pro of of the lemma, will be a simple cons equence from the follo wing series of claims . W e stress that throughout we will not compute β explicitly , but we will assume it is small enough as a function of ε for the arguments to go through. Claim 4.6. W.h.p. every U ⊆ V of c ar dinality | U | ≤ n 0 . 11 ln n satisfies e G 1 − H 1 ( U ) ≤ e G ( U ) ≤ ( np ) 3 / 25 | U | . Pr o of. Fixing such a subset of vertices U o f cardinality u ≤ n 0 . 11 ln n , we ha ve that e G ( U ) ∼ Bin  u 2  , p  and s ince e  u 2  p ≤ u ( np ) 3 / 25 · n − 1 / 200 we hav e by Lemma 2.3 that Pr h e G ( U ) > ( np ) 3 / 25 u i < e  u 2  p ( np ) 3 / 25 u ! ( np ) 3 / 25 u ≤ exp  − ( np ) 3 / 25 · u · ln n 200  ≤ exp  − (ln n ) 1 . 12 · u 200  . T o upper bound the probability of the existence of a subset of vertices for which the ass ertion of the claim do es no t hold, we apply the union b ound over all p ossible sets U of car dinality u ≤ n 0 . 11 ln n n 0 . 11 ln n X u =1  n u  exp  − (ln n ) 1 . 12 · u 200  ≤ n 0 . 11 ln n X u =1 exp  u ·  ln en u − (ln n ) 1 . 12 200  ≤ n 0 . 11 ln n · exp  − (ln n ) 1 . 1  = o (1) . 14 Claim 4.7. W.h.p. every U ⊆ V of c ar dinality | U | ≤ 12 β n satisfies e G 1 − H 1 ( U ) ≤ e G ( U ) ≤ 5 0 β np | U | . Pr o of. V ery similarly to the pro of of Claim 4 .6 w e fix a subse t of v e rtices U of cardina lit y 1 ≤ u ≤ 12 β n , then noting tha t e G ( U ) ∼ Bi n  u 2  , p  and s ince  u 2  p ≤ 50 β n p u we hav e by Lemma 2.3 that Pr [ e G ( U ) > 5 0 β npu ] < e  u 2  p 50 β npu ! 50 β npu ≤ exp  − 50 β npu ln 100 β n eu  . T o upper bound the probability of the existence of a subset of vertices for which the ass ertion of the claim do es no t hold, we apply the union b ound over all p ossible sets U of car dinality 1 ≤ u ≤ 12 β n 12 β n X u =1  n u  exp  − 50 β npu ln 100 β n eu  ≤ 12 β n X u =1 exp  u ·  ln en u − 50 β np ln 100 β n eu  ≤ 12 β n X u =1 exp  u ·  ln n u (1 − 5 0 β ln n ) + 1 − 50 β ln n ln 100 β e  = o (1) . Claim 4.8. W.h.p. every two disjoint subsets U, Z ⊆ V 1 wher e | U | = β n 1 and | Z | = n 1  1 3 − 27 β  satisfy e G 1 − H 1 ( U, Z ) ≥ n 1 ln ln n 1 . Pr o of. Fix t wo such disjoint subsets of vertices U, Z ⊆ V 1 of the required c ardinalities and no te that e G 1 − H 1 ( U, Z ) = e G 1 ( U, Z ) − e H 1 ( U, Z ) ≥ e G ( U, Z ) − X v ∈ U d H 1 ( v ) ≥ e G ( U, Z ) −  1 3 − ε  X v ∈ U d G ( v ) . As e G ( U, Z ) ∼ Bin ( | U | · | Z | , p ), we can apply Theor em 2.2 item 2 and get that Pr  e G ( U, Z ) <  1 3 − 28 β  n 1 · | U | · p  ≤ e − Θ( n 2 1 p ) = o (4 − n 1 ) , where the hidden cons tant s in the exp onent ab ov e are functions of β alo ne. Next, w e prove that the ra ndom v ariable X ( G ) = P v ∈ U d G ( v ) is very lik ely not to devia te m uch from its expecta tion. T o ac hieve this we resort to the Azuma-Ho effding inequality for martinga les of bo unded v ariance (see e.g. [1, Theorem 7.4.3]). Note tha t for an y tw o graphs on the vertex set V that differ by a single edge, their v alue of X can change by at most 2, and E [ X ( G )] = | U | ( n − 1) p < (1 + o (1)) | U | n 1 p . In order to divulg e the v alue of X one only needs to expose the pair s o f v ertices which hav e at least one endpoint in U . This implies that the total v ariance of the martingale is at mos t β n 2 1 p (1 − p ), and hence Pr [ X ( G ) > (1 + β ) | U | n 1 p ] ≤ e − Θ( n 2 1 p ) = o (4 − n 1 ) , where the hidden c onstants in the exp onent a bove are, a gain, functions of β a lone. Rec alling that β can b e made s mall enough with resp ect to ε and that | U | n 1 p ≫ n 1 ln ln n 1 we hav e that  1 3 − 28 β  >  1 3 − ε  (1 + β ) + n 1 ln ln n 1 | U | n 1 p , and hence Pr [ e G 1 − H 1 ( U, Z ) < n 1 ln ln n 1 ] = o (4 − n 1 ). By applying t he union b ound ov er all pairs of subsets of vertices U and Z , the pro of of the claim in co mpleted. 15 Recalling Definition 3.1 completes the pro of of the lemma. W e can now present the main r esult o f this section which ba sed on the ab ov e is readily es tablished. In what follows let G = ( V , E ) with vertex set V = [ n ] b e sampled from G ( n, p ) a nd let t = np 100 . W e recall that for every ε > 0 if d = ( d 1 , . . . , d n ) is the degree sequence of G , then e d ( t, ε ) = ( e d 1 , . . . , e d n ) is the sequence defined b y e d v = d v − 2 for every v ∈ D t ( G ) and e d v = d v (1 / 3 − ε ) for every V 1 = V \ D t ( G ). Corollary 4.9. F or every fixe d ε > 0 , ther e exists a smal l enough c onstant β 0 = β 0 ( ε ) > 0 such that for every 0 < β ≤ β 0 and p ≥ ln n +ln ln n + ω (1) n the fol lowing holds. W.h.p. every sub gr aph H ⊆ G of de gr e e se quenc e d H ≤ e d ( t, ε ) is such that the gr aph G − H c ontains a sub gr aph Γ 0 ∈ L ( n, β ) which sp ans at most 2 · 10 6 β n 2 p e dges. Mor e over, addi ng to Γ 0 any su bset of e dges E 0 ⊆ E r esults in a gr aph in L ( n, β ) and t he p artition V = D t ( G ) ∪ V 1 guar ante es that Γ 0 + E 0 is in L ( n, β ) . Pr o of. Fix ε > 0, le t β 0 be as g uaranteed b y Lemma 4.5, let 0 < β ≤ β 0 , a nd se t t = np 100 . Fix a subgraph H ⊆ G with deg ree sequence d H ≤ e d ( t, ε ), denote b y G 1 = G 1 ( t ) = ( V 1 , E 1 ), let H 1 = H [ V 1 ], and set n 1 = | V 1 | . W e can a ssume tha t G satisfies the following pro per ties: 1) δ ( G − H ) ≥ 2 (Theorem 4.2). 2) |D t ( G ) | ≤ n 0 . 09 (Claim 4.3). 3) There is no path o f length at mo st 2 ln n 3 ln ln n with b oth (p ossibly identical) endp oints in |D t ( G ) | (Claim 4.4), 4) G 1 − H 1 contains a ( β n 1 , f β )-expander spanning subgraph Γ with at most 10 6 β n 2 p edges (Lemmata 4.5 a nd 3 .2 where Cla im 4.1 can b e used to bound the num ber of edges in e ( G 1 )). T o ge t the graph Γ 0 , we a dd to the graph Γ the set of v ertices D = D t ( G ) with all of its incident edges from G − H . Note that Γ 0 ∈ L ( n, β ) (using the partition V = V 1 ∪ D ) a nd that e (Γ 0 ) = e (Γ) + e G − H ( D , V 1 ) ≤ 10 6 β n 2 p + t · n 0 . 1 < 2 · 10 6 β n 2 p a s cla imed. Consider an y subset o f edges E 0 ⊆ E ( G ). As D is independent, a ll edg es of E 0 which are not in Γ 0 m ust hav e at lea st one endp oint in V 1 . The addition cannot create a path of length at most 2 ln n 3 ln ln n betw een endpo in ts in D since no such path exists in G . The addition of any edge from E G ( D , V 1 ) to Γ 0 can only increase the degr ee of every vertex fro m D , and, fina lly , the addition of any edge from E G ( V 1 ) to Γ 0 clearly leav e s t he induced subgraph on V 1 as a ( β n 1 , f β )-expander (as this is a monoto ne increasing graph property), and ther efore the same pa rtition o f the vertex se t V = V 1 ∪ D also implies that Γ 0 + E 0 ∈ L ( n, β ). 5 Pro of of Theorem 1 W e can no w provide the full pro of of the main result o f this paper , na mely the pro of of Theorem 1. The road w e tak e to achiev e this is to s how that g iven a typical graph G sampled from G ( n, p ), no matter how H ⊆ G is chosen (giv en it satisfies the conditions on its degre e sequence), not only will the gra ph G − H contain a sparse expander s ubgraph Γ, it will als o hav e as edges enough b o osters with resp ect to Γ as to transform it into a Hamiltonian g raph. 16 Pr o of of The or em 1. Fix ε > 0, s et β = β ( ε ) to b e a sufficiently s mall constant such that the assertion of Corollar y 4.9 ho lds, and let G ∼ G ( n, p ) be with deg ree sequence d . If G does not satisfy the conclusion of Corollar y 4.9 w e say that G is c orrupte d , and denote by A G this even t of probability o (1). Assume, then, that G is not co rrupted and not e d = e d ( np 100 , ε )-r esilient to H AM , i.e. there exists a subgraph H 0 with degree sequence d H ≤ e d for whic h G − H 0 is not Hamiltonian. Coro llary 4.9 implies G − H 0 contains a subgraph Γ 0 ∈ L ( n, β ) whic h spans at mos t 2 · 10 6 β n 2 p edges , and mo reov er, a dding to it any subset of edg es from E 1 ⊆ E ( G ), results in a graph Γ 0 + E 1 ∈ L ( n, β ). Lemma 3 .4 implies that there m ust exist a set E 0 ⊆ E ( G − H ) ⊆ E ( G ) of at most n edges for which | N G − (Γ 0 + E 0 ) ( v ) ∩ B Γ 0 + E 0 ( v ) | ≤ e d v for every vertex v ∈ V . As | E 0 | ≪ e (Γ 0 ) from the a bove we co nclude that Γ 2 = Γ 0 + E 0 ∈ L ′ ( n, β ) = { Γ ∈ L ( n, β ) : | E (Γ) | ≤ 1 0 7 β n 2 p } . Corollar y 4.9 guarantees that V = V 1 ∪ D t ( G ) is a partition of the vertex se t for which Γ 2 satisfies the prop erties o f L ( n, β ). Lemma 3 .6 implies the set A = { v ∈ V 1 : | B Γ 2 ( v ) | ≥ n (1 / 3 + β ) } m ust satisfy | A | ≥ n (1 / 3 + β ). Let A 0 ⊆ A b e a subset o f cardinality | A 0 | = n 3 . So, in fact, for non-cor rupted G we will reso rt to b ound the even t that there exists a gr aph Γ 2 ∈ L ′ that is contained in G fo r which | N G − Γ 2 ( v ) ∩ B Γ 2 ( v ) | ≤ e d v =  1 3 − ε  d v for every vertex v ∈ A 0 . One should note that from the indep endence of the app earance of edges in the G ( n, p ) mo del, given some Γ 2 ∈ L ′ the even ts [Γ 2 ⊆ G ] a nd h | N G − Γ 2 ( v ) ∩ B Γ 2 ( v ) | ≤ e d v i are independent as they stem from the app earanc e of disjoint sets of edg es, i.e. e (Γ 2 ) and  V 2  \ e (Γ 2 ) resp ectively . Putting it all together yields that the proba bilit y that G is not e d -resilient to HAM is uppe r bo unded b y Pr [ A G ] + Pr h ∃ Γ 2 ∈ L ′ . ( Γ 2 ⊆ G ) ∧  ∀ v ∈ V . | N G − Γ 2 ( v ) ∩ B Γ 2 ( v ) | ≤ e d v     A G i ≤ o (1) + 1 Pr  A G  X Γ 2 ∈L ′ Pr [Γ 2 ⊆ G ] · Pr h ∀ v ∈ V . | N G − Γ 2 ( v ) ∩ B Γ 2 ( v ) | ≤ e d v i ≤ o (1) + (1 + o (1)) X Γ 2 ∈L ′ p e (Γ 2 ) × Pr " X v ∈ A 0 | N G − Γ 2 ( v ) ∩ B Γ 2 ( v ) | ≤  1 3 − ε  (2 e G ( A 0 ) + e G ( A 0 , V \ A 0 )) # . (4) T aking into ac count that w e are using a union b ound argument by summing ov er all graphs Γ 2 ∈ L ′ (and there may b e an exp onential n umber o f those) it is le ft to show that every s ummand in the ab ov e expressio n is ex po nen tially small. W e define the following r andom v ariable (which depends o n the c hoice of Γ 2 ). Let X = X ( G ) = X v ∈ A 0 | N G − Γ 2 ( v ) ∩ B Γ 2 ( v ) | , whose exp ectation s atisfies E [ X ] = X v ∈ A 0 E [ | N G − Γ 2 ( v ) ∩ B Γ 2 ( v ) | ] = p · X v ∈ A 0 E [ | B Γ 2 ( v ) | ] ≥ n 2 p 9 . Note tha t for an y tw o graphs on the vertex set V that differ by a single edge, their v alue of X can change by at most 2 (1 for every endpoint of the e dge), hence w e can apply the Azuma-Ho effding inequalit y for martingales of bounded v a riance (see e.g. [1, Theorem 7.4.3]), to prov e that X is concentrated around its exp ectation. In the pro cess o f “exp osing” the edges of the g raph, it suffices to e xpo se only the pairs with an endpo in t in A 0 which are non-edges of Γ 2 . This implies that the total v ariance o f the mar tingale is upp er 17 bo unded b y n 2 3 p (1 − p ), and hence Pr  X ( G ) ≤ n 2 p 9 (1 − ε )  ≤ exp  − Θ( ε 2 n 2 p )  . (5) On the o ther ha nd, a sta ndard a pplication of Theor em 2 .2 (the Cherno ff b ound) it follows that Pr  ∃ U ⊆ V . | U | = n 3 ∧ 2 e ( U ) + e ( U, V \ U ) > n 2 p 3 (1 + ε )  = exp  − Θ( ε 2 n 2 p )  . (6) Using the fac t tha t n 2 p 9 (1 − ε ) ≥ n 2 p 3 ( 1 3 − ε )(1 + ε ), it follows from (5) and (6) that Pr  X ≤  1 3 − ε  (2 e G ( A 0 ) + e G ( A 0 , V \ A 0 ))  ≤ exp  − Θ( ε 2 n 2 p )  . Returning to our upper b ound on the probability that G is no t e d -resilient to HAM , w e set µ = 10 7 β . Plugging in the above in (4) the pro babilit y is upp er b ounded by o (1) + (1 + o (1)) µn 2 p X m =1   n 2  m  · p m · exp( − Θ( ε 2 n 2 p )) ≤ o (1) + (1 + o (1)) µn 2 p X m =1  en 2 p 2 m  m · exp( − Θ( ε 2 n 2 p )) ≤ o (1) + exp(Θ  µn 2 p ln 1 µ  − Θ( ε 2 n 2 p )) = o (1) , where from the second to the second inequality follows fro m the fact that  en 2 p m  m is increa sing with m for the given range, and that β (a nd µ ) can be chosen small enough with resp ect to ε . This completes the pro of of the theorem. 6 Concluding remarks and further researc h directions This work is y et another building blo c k in the recently initiated research a rea of resilience of gra ph prop- erties. The gener alized appr oach allo wed us to tackle in a uniform w ay t wo different pr oblems reg arding Hamiltonicity . The main motiv atio n for studying resilience o f the type consider ed in this w ork is the ability to hav e refined control ov er vertices of small degr ee which create the main obstacle for Hamiltonicity when p is in the low end of the range. As p g rows the degree sequence of the graph be comes more and more balanced, so this appro ach do es not seem to be suitable (o r even necess ary) for la rger v alues o f p . 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