On the Chromatic Number of Random Graphs with a Fixed Degree Sequence

Let d = 1 ≤ d1 ≤ d2 ≤ · · · ≤ dn be a non-decreasing sequence of n positive integers, whose sum is even. Let Gn,d denote the set of graphs with vertex set [n] = {1, 2, . . . , n} in which the degree of vertex i is di. Let Gn,d be chosen uniformly at random from Gn,d. Let d = (d1 + d2 + · · ·+ dn)/n be the average degree. We give a condition on d under which we can show that whp the chromatic number of Gn,d is Θ(d/ ln d). This condition is satisfied by graphs with exponential tails as well those with power law tails.