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arXiv · 2401.14590

The forb-flex method for odd coloring and proper conflict-free coloring of planar graphs

James Anderson·Herman Chau·Eun-Kyung Cho·Nicholas Crawford·Stephen G. Hartke·Emily Heath·Owen Henderschedt·Hyemin Kwon·Zhiyuan Zhang
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Abstract

We introduce a new tool useful for greedy coloring, which we call the forb-flex method, and apply it to odd coloring and proper conflict-free coloring of planar graphs. The odd chromatic number, denoted $\chi_{\mathsf{o}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing an odd number of times in its neighborhood. The proper conflict-free chromatic number, denoted $\chi_{\mathsf{PCF}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing uniquely in its neighborhood. Our new tool works by carefully counting the structures in the neighborhood of a vertex and determining if a neighbor of a vertex can be recolored at the end of a greedy coloring process to avoid conflicts. Combining this with the discharging method allows us to prove $\chi_{\mathsf{PCF}}(G) \leq 4$ for planar graphs of girth at least 11, and $\chi_{\mathsf{o}}(G) \leq 4$ for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park.

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James Anderson, Herman Chau, Eun-Kyung Cho, Nicholas Crawford, Stephen G. Hartke, Emily Heath, Owen Henderschedt, Hyemin Kwon, Zhiyuan Zhang. 2024-01-26. The forb-flex method for odd coloring and proper conflict-free coloring of planar graphs. https://arxiv.org/abs/2401.14590

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