Bounds for the game coloring number of planar graphs with a specific girth
Abstract
Let colg(G) be the game coloring number of a given graph G. Define the game coloring number of a family of graphs H as colg(H) := \ colg(G):G ∈ H\. Let Pk be the family of planar graphs of girth at least k. We show that colg(P7) ≤ 5. This result extends a result about the coloring number by Wang and Zhang WZ11 ( colg(P8) ≤ 5). We also show that these bounds are sharp by constructing a graph G where G ∈ colg(Pk) ≥ 5 for each k ≤ 8 such that colg(G)=5. As a consequence, colg(Pk) = 5 for k =7,8.
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