On the chromatic number of random d-regular graphs

Abstract

In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d < 2(k − 1) log(k − 1). From previous lower bounds due to Molloy and Reed, this establishes the chromatic number to be asymptotically almost surely k − 1 or k. If moreover d > (2k − 3) log(k − 1), then the value k − 1 is discarded and thus the chromatic number is exactly determined. Hence we improve a recently announced result by Achlioptas and Moore in which the chromatic number was allowed to take the value k + 1. Our proof applies the small subgraph conditioning method to the number of balanced k-colourings, where a colouring is balanced if the number of vertices of each colour is equal.celebrated result by Bollobás [7] later extended by Luczak [14] showed that if pn → ∞ then asymptotically almost surely (a.a.s.) χ(G(n, p)) ∼ n log(1/(1 − p)) 2 log(np) .Here and in similar statements, an event occurs a.a.s. if its probability tends to 1 as n tends to infinity. For p = c/n, Achlioptas and Naor [3] proved that the chromatic number of G(n, p) is a.a.s. k or k + 1 where k is the smallest positive integer with 2k log k > c. Moreover, they discarded the case k for roughly half of the values of c. In the same direction, Coja-Oghlan, Panagiotou and Steger [8] showed that a.a.s. χ(G(n, p)) ∈ {k, k + 1, k + 2} for p < n −3/4−ǫ where k is the smallest positive integer satisfying 2k log k > p(n − 1). Meanwhile, some other results gave concentration of the chromatic number without determining the values so precisely: Luczak [15] proved that χ(G(n, p)) is a.a.s. 2-point-concentrated if p < n −5/6−ǫ , and later Alon and Krivelevich [4] extended this to p < n −1/2−ǫ . More recently, results have been published about the chromatic number for the model G n,d of random d-regular graphs, which is the probability space on d-regular graphs with n vertices having uniform distribution. For basic results and notation on random regular graphs, see [21]. Hereinafter, dn is always assumed to be even for feasibility. For fixed d, Molloy and Reed [16] showed that if q(1 − 1/q) d/2 < 1 then χ(G n,d ) > q a.a.s. Then, for d < n 1/3−ǫ , Frieze and Luczak [11] established thatand later Cooper, Frieze, Reed and Riordan [9] extended the same asymptotic formula to apply to d ≤ n 1−ǫ . Similarly, the range n 6/7+ǫ ≤ d ≤ 0.9n was covered by Krivelevich, Sudakov, Vu and Wormald [13], who showed that χ(G n,d ) ∼ n 2 log b d a.a.s. where b = n/(n − d). Achlioptas and Moore [2] recently announced a significant new result for constant d. They stated that if k is the smallest integer satisfying d < 2(k − 1) log(k − 1) then a.a.s. χ(G n,d ) is k − 1, k, or k + 1. If, in addition, d > (2k − 3) log(k − 1), then a.a.s. χ(G n,d ) is k or k + 1. Finally, Ben-Shimon and Krivelevich [5] established 2-point concentration of χ(G n,d ) for d = o(n 1/5 ).In this paper we restrict the set of possible values for the chromatic number given by Achlioptas and Moore, and show that χ(G n,d ) a.a.s. ca...

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