On graphs with maximum difference between game chromatic number and chromatic number

TL;DR

This paper investigates the maximum difference between the game chromatic number and the chromatic number of a graph, confirming a conjecture that only specific small graphs and certain Turán graphs achieve equality.

Contribution

The paper proves Matsumoto's conjecture by characterizing graphs where the maximum difference occurs, including a modified game variant and a counterexample for the marking game.

Findings

Confirmed Matsumoto's conjecture on maximum difference cases

Characterized graphs achieving the maximum difference

Provided a counterexample for the vertex marking game

Abstract

In the vertex colouring game on a graph $G$, Maker and Breaker alternately colour vertices of $G$ from a palette of $k$ colours, with no two adjacent vertices allowed the same colour. Maker seeks to colour the whole graph while Breaker seeks to make some vertex impossible to colour. The game chromatic number of $G$, $\chi_g(G)$, is the minimal number $k$ of colours for which Maker has a winning strategy for the vertex colouring game. Matsumoto proved in 2019 that $\chi_g(G)-\chi(G)\leq\lfloor n/2\rfloor - 1$, and conjectured that the only equality cases are some graphs of small order and the Tur\'{a}n graph $T(2r,r)$ (i.e. $K_{2r}$ minus a perfect matching). We resolve this conjecture in the affirmative by considering a modification of the vertex colouring game wherein Breaker may remove a vertex instead of colouring it. Matsumoto further asked whether a similar result could be proved for the vertex marking game, and we provide an example to show that no such nontrivial result can exist.