Let G(n,d) be the random d-regular graph on n vertices. For any integer k exceeding a certain constant k_0 we identify a number d_ k-col such that G(n,d) is k-colorable w.h.p. if d d_ k-col .
With respect to random regular graphs @math , Frieze and @cite_20 proved a result akin to 's @cite_14 for @math . In fact, Cooper, Frieze, Reed and Riordan @cite_5 extended this result to the regime @math for any fixed @math , and Krivelevich, Sudakov, Vu and Wormald @cite_33 further still to @math . For @math fixed as @math , the bounds from @cite_20 were improved by the aforementioned contributions @cite_18 @cite_17 .
@cite_18 @cite_14 @cite_33 @cite_5 @cite_20 @cite_17
2165387581 2012724629 2139757566 2018483383
Given any integer d ≥ 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2. Let χ(G(n, p)) denote the chromatic number of the random graphG(n, p). We prove that there exists a constantd0 such that fornp(n)>d0,p(n)→0, the probability that @math tends to 1 asn→∞. Let r = r(n) → ∞ with 3 l r l n1−η for an arbitrarily small constant η > 0, and let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set l1, 2, …, nr. We prove that, with probability tending to 1 as n → ∞, Gr has the following properties: the independence number of Gr is asymptotically 2n log r r and the chromatic number of Gr is asymptotically r 2nlogr. Abstract Let G r denote a random r -regular graph with vertex set 1, 2, …, n and α ( G r ) and χ ( G r ) denote respectively its independence and chromatic numbers. We show that with probability going to 1 as n → ∞ respectively |δ(G r ) − 2n r ( log r − log log r + 1 − log 2)|⩽ γn r and |χ(G r ) − r 2 log r − 8r log log r ( log ) 2 | ⩽ 8r log log r ( log r ) 2 provided r = o ( n θ ), θ 1 3 , 0 r ≥ r e , where r e depends on e only.