The game of arboricity
Tomasz Bartnicki, Jarosław Grytczuk, Hal Kierstead
Abstract
Using a fixed set of colors C, Ann and Ben color the edges of a graph G so that no monochromatic cycle may appear. Ann wins if all edges of G have been colored, while Ben wins if completing a coloring is not possible. The minimum size of C for which Ann has a winning strategy is called the game arboricity of G, denoted by Ag(G). We prove that Ag(G) ≤3k for any graph G of arboricity k, and that there are graphs such that Ag(G)≥2k-2. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction.
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