The Game Chromatic Number of Dense Random Graphs

Abstract

Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all vertices are colored. The game chromatic number χ_g(G) is defined as the smallest k for which the first player has a winning strategy.

Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In this paper we improve existing results and show that with high probability, the game chromatic number χ_g(G_n,p) of dense random graphs with p ≥ e^{-o(log n)} is asymptotically twice as large as the ordinary chromatic number χ(G_n,p).