The coloring game on matroids

TL;DR

This paper introduces a game-theoretic variant of matroid coloring, establishing an upper bound of twice the chromatic number for the game chromatic number, which is nearly tight based on constructed examples.

Contribution

It proves that the game chromatic number of any matroid is at most twice its chromatic number, extending previous results from graphic to general matroids.

Findings

Proved  bound for all matroids.

Constructed matroids with  tightness.

Extended previous graphic matroid results.

Abstract

A coloring of the ground set of a matroid is proper if elements of the same color form an independent set. For a loopless matroid $M$, its chromatic number $\chi (M)$ is the minimum number of colors in a proper coloring. In this note we study a game-theoretic variant of this parameter. Suppose that Alice and Bob alternately properly color the ground set of a matroid $M$ using a fixed set of colors. The game ends when the whole matroid has been colored, or if they arrive to a partial coloring that cannot be further properly extended. Alice wins in the first case, while Bob in the second. The game chromatic number of $M$, denoted by $\chi_{g}(M)$, is the minimum size of the set of colors for which Alice has a winning strategy. Clearly, $\chi_{g}(M)\geq\chi (M)$. We prove an upper bound $\chi_{g}(M)\leq 2\chi (M)$ for every matroid $M$. This improves and extends a result of Bartnicki, Grytczuk and Kierstead, who showed that $\chi_{g}(M)\leq 3\chi (M)$ holds for graphic matroids. Our bound is almost tight, as we construct a family of matroids $M_k$ (for $k\geq 3$) satisfying $\chi (M_k)=k$ and $\chi_{g}(M_k)=2k-1$.