Let ${ \rm col_g}(G)$ be the game coloring number of a given graph $G.$ Define the game coloring number of a family of graphs $\mathcal{ H}$ as ${ \rm col_g}(\mathcal{ H}) := \max\{ { \rm col_g}(G):G \in \mathcal{ H}\}.$ Let $\mathcal{ P}_k$ be the family of planar graphs of girth at least $k.$ We show that ${ \rm col_g}(\mathcal{ P}_7) \leq 5.$ This result extends a result about the coloring number by Wang and Zhang { WZ11} (${ \rm col_g}(\mathcal{ P}_8) \leq 5).$ We also show that these bounds are sharp by constructing a graph $G$ where $G \in { \rm col_g}(\mathcal{ P}_k) \geq 5$ for each $k \leq 8$ such that ${ \rm col_g}(G)=5.$ As a consequence, ${ \rm col_g}(\mathcal{ P}_k) = 5$ for $k =7,8.$
Categories: math.CO