# The Game Coloring Number Of Planar Graphs With A Specific Girth

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The game coloring number of planar graphs with a specific girth. By keaitsuda maneeruk nakprasit and kittikorn nakprasit. This implies that the game chromatic number and the game coloring number of every planar graph with girth at least 9 is at most 5. Coloring game problems arose as game theoretic versions of well known graph coloring problems. We prove that colg g colg g is at most 13 if gg is a planar graph with girth at least 4.

When used without any qualification a coloring of a graph is almost always a proper vertex coloring namely a labeling of the graph s vertices with colors such that no two vertices sharing the same edge have the same color. We denote by χ g g the game chromatic number of a graph g which is the smallest number of colors alice needs to win the coloring game on g we know from montassier et al. 2012 and independently from wang and zhang 2011 that planar graphs with girth at least 8 have game chromatic number at most 5. Get pdf 94 kb.

A connection directly back to itself could never be properly colored it is understood that graphs in this context are. Let col g g be the game coloring of a given graph g. Define the game coloring number of a family of graphs mathcal h as mathrm col g mathcal h max mathrm col g g g in mathcal h let mathcal p k be the family of planar graphs of girth at least k. Bounds for the game coloring number of planar graphs with a specific girth.

Let mathrm col g g be the game coloring number of a given graph g. We show that col g g 12 when g is a planar graph with girth at least 4 and col g g 5 when g is a planar graph with girth at least 7. We also show that there is a planar graph gg with girth 4 such that colg g 7colg g 7 and there is a. In this paper we prove that the game coloring number of planar graphs with girth 4 is at most 13 by constructing an appropriate linear ordering.

In a coloring game two players use a given set of colors to construct a coloring of a graph following specific rules depending on the game we consider one player tries to successfully complete the coloring of the graph when. By an fm coloring of a graph we mean a partition of its edges into a forest colored with and a matching colored with. Bounds for the game coloring number of planar graphs with a specific girth.

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