On the game chromatic number of sparse random graphs

TL;DR

This paper investigates the game chromatic number of sparse random graphs and regular graphs, analyzing how the number of colors needed for a winning strategy behaves in these cases.

Contribution

It extends previous work by analyzing the asymptotic behavior of the game chromatic number for graphs with constant average degree and for random regular graphs.

Findings

Provides bounds for the game chromatic number in sparse graphs

Analyzes the behavior for random regular graphs

Extends understanding of the parameter in low-density 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. The paper \cite{BFS} 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.