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Random Regular Graphs of Non-Constant Degree: Independence and Chromatic Number

Let r = r(n) → ∞ with 3 [les ] r [les ] 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 {1, 2, …, n}. 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.