The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses

TL;DR

This paper investigates the game chromatic number of planar graphs with large girth, proving the bound of 5 cannot be improved and constructing cactuses with large girth that still require 5 colors to win.

Contribution

It demonstrates that the known upper bound of 5 for planar graphs with large girth is tight by constructing specific cactuses with large girth that also require 5 colors.

Findings

Planar graphs with girth at least 8 have game chromatic number at most 5.

The bound of 5 cannot be improved for large girth graphs.

Existence of cactuses with large girth and game chromatic number exactly 5.

Abstract

We denote by $\chi$ 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. [M. Montassier, P. Ossona de Mendez, A. Raspaud and X. Zhu, Decomposing a graph into forests, J. Graph Theory Ser. B, 102(1):38-52, 2012] and, independantly, from Wang and Zhang, [Y. Wang and Q. Zhang. Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that planar graphs with girth at least 8 have game chromatic number at most 5. One can ask if this bound of 5 can be improved for a sufficiently large girth. In this paper, we prove that it cannot. More than that, we prove that there are cactuses CT (i.e. graphs whose edges only belong to at most one cycle each) having $\chi$ g (CT) = 5 despite having arbitrary large girth, and even arbitrary large distance between its cycles.