The game chromatic number of random graphs

TL;DR

This paper investigates the asymptotic behavior of the game chromatic number in random graphs G_{n,p}, showing it is roughly twice the standard chromatic number with high probability, and extends analysis to bipartite graphs.

Contribution

It provides the first asymptotic analysis of the game chromatic number in random graphs and bipartite graphs, establishing bounds relative to the standard chromatic number.

Findings

Game chromatic number is at least twice the chromatic number with high probability.

Game chromatic number has the same order of magnitude as the chromatic number, up to a constant.

Analysis extends to random bipartite graphs.

Abstract

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 \chi_g(G) is the minimum k for which the first player has a winning strategy. In this paper we analyze the asymptotic behavior of this parameter for a random graph G_{n,p}. We show that with high probability the game chromatic number of G_{n,p} is at least twice its chromatic number but, up to a multiplicative constant, has the same order of magnitude. We also study the game chromatic number of random bipartite graphs.