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arXiv · 1211.2456

The coloring game on matroids

Michał Lasoń
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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 $χ(M)$ 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 $χ_{g}(M)$, is the minimum size of the set of colors for which Alice has a winning strategy. Clearly, $χ_{g}(M)\geqχ(M)$. We prove an upper bound $χ_{g}(M)\leq 2χ(M)$ for every matroid $M$. This improves and extends a result of Bartnicki, Grytczuk and Kierstead, who showed that $χ_{g}(M)\leq 3χ(M)$ holds for graphic matroids. Our bound is almost tight, as we construct a family of matroids $M_k$ (for $k\geq 3$) satisfying $χ(M_k)=k$ and $χ_{g}(M_k)=2k-1$.

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Michał Lasoń. 2014-12-26. The coloring game on matroids. https://doi.org/10.1016/j.disc.2016.11.020

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