Canonical labelling of random regular graphs

Given an input graph G, a canonical labelling algorithm computes a bijection π G : V (G) → {1, . . . , n} with the following property: if a graph G ′ is isomorphic to G, then the relabelled versions of G and G ′ , under the actions of π G and π G ′ , are identical. There is a linear time reduction from the graph isomorphism problem to canonical labelling: once the labellings π G , π G ′ have been computed for two input graphs G, G ′ , it takes time O(|E(G)|) to check whether G, G ′ are isomorphic. The best 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-vertex graphs. A quasi-polynomial bound exp(log O (1) n) is also known for the canonical labelling problem [8]. Nevertheless, for graphs with bounded degree, both problems can be solved in polynomial time [10,40]. In particular, this is the case for d-regular graphs when d = const. In this paper, we show that there exists a polynomial-time canonical labelling algorithm for almost all d-regular graphs for all 0 ≤ d ≤ n -1.

Colour refinement (CR) is a simple algorithmic routine that operates on vertex-coloured graphs. For an input graph G with initial colouring C 0 : V (G) → Z, CR iteratively computes new colourings. At 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 multiset of C t-1 -colours of neighbours of v. That is, the process refines the initial partition C 0 and halts once the partition stabilises. Let us call a colouring discrete if every pair of vertices 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 vertex colours are isomorphism-invariant, this yields a canonical labelling by numbering the colour names in the lexicographic order. In [9,19,26], it was proved that CR run on a binomial random graph G(n, p) 1 outputs a discrete colouring with high probability (whp, in what follows) 2 whenever (1 + ε) ln n n < p ≤ 1 2 , which implies a near linear time algorithm 3 for canonical labelling of G(n, p) [13].

We stress that for regular uncoloured graphs G, CR terminates immediately with a trivial colouring, and is therefore unsuitable for canonical labelling. For a positive integer n, we denote [n] := {1, . . . , n}. Let d ≤ n -1 be a non-negative integer such that dn is even. Let G n,d be a uniform distribution over all d-regular graphs on [n]. We 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, efficient canonical labelling algorithms are known, we focus on the case d = ω (1). Moreover, since the edge complement of a d-regular graph is (n -1 -d)-regular, the edge complement of G n,n-1-d is distributed as G n,d . Therefore, we may restrict ourselves to d ≤ n/2. Our main result shows that the triviality of the initial colouring is the only obstacle for a complete refinement: once a non-trivial initial colouring is produced, whp CR run on G n ∼ G n,d outputs a discrete colouring which is suitable for canonical labelling.

Before we state the main result of this paper, we need one more definition. For a connected graph G, we denote by diam(G) its diameter.

Theorem 1. Let d 0 be large enough, let d = d(n) be such that d 0 ≤ d ≤ n/2, and let G n ∼ G n,d . Then, the following holds whp: for every non-trivial partition [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 discrete colouring.

Remark 2. We did not try to optimise the bound on the number of rounds, and we believe that 2 diam(G n ) + 3 is suboptimal. 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 suspect that the total number of rounds needed is (1 + o diam(Gn) (1)) diam(G n ) whp. We also note that our technique does not directly generalise to small values of d; in particular, there appears to be a natural barrier at d = 3. Nevertheless, we believe that the statement of Theorem 1 can be extended to all 3 ≤ d ≤ n/2.

We now describe a possible approach to canonical labelling of G n ∼ G n,d based on our main result. Since the output of CR can be 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]. To this end, recall that, for any d = ω(1), whp G n contains a triangle [32]. Let t i be the number of triangles that contain the vertex i ∈ [n] in G n , and let t = max{t 1 , . . . , t n }. Using fast square matrix multiplication, it is possible to compute the vector (t 1 , . . . , t n ) in time n ω , where ω < 2.372 [1]. Alternatively, this vector can be computed in time O(nd 2 ) using the standard triangle-listing algorithm [17], which is faster when d < n 0.685 , assuming the best known upper bound on the matrix multiplication exponent ω (and yields the bound o(n 2 ) for

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