On the game chromatic number of sparse random graphs

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χg(G) is the minimum k for which the first player has a winning strategy. The paper [T. Bohman, A. M. Frieze, and B. Sudakov, Random Structures Algorithms, 32 (2008), pp. 223-235] began the analysis of the asymptotic behavior of this parameter for a random graph G_n,p. This paper provides some further analysis for graphs with constant average degree, i.e., np = O(1), and for random regular graphs. We show that with high probability (w.h.p.) c₁χ(G_n,p) ≤ χg(G_n,p) ≤ c₂χ(Gn,p) for some absolute constants 1 < c₁ < c₂. We also prove that if G _n,3 denotes a random n-vertex cubic graph, then w.h.p. χg(G _n,3) = 4.