The game chromatic number of dense random graphs

TL;DR

This paper proves that in dense random graphs, the game chromatic number is asymptotically twice the usual chromatic number, refining previous bounds and providing a precise asymptotic relationship.

Contribution

It establishes that for dense random graphs, the game chromatic number is asymptotically twice the chromatic number, improving earlier bounds and deepening understanding of the game coloring dynamics.

Findings

Game chromatic number is asymptotically twice the chromatic number in dense graphs.

Improved bounds on the game chromatic number for random graphs.

High probability results for dense random graph coloring strategies.

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 the vertices are colored. The game chromatic number $\chi_g(G)$ 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 $\chi_g(G_{n,p})$ of dense random graphs is asymptotically twice as large as the ordinary chromatic number $\chi(G_{n,p})$.