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arXiv · 2205.09856

List Multicoloring of Planar Graphs and Related Classes

Glenn G. Chappell
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Abstract

For positive integers $a$ and $b$, a graph $G$ is $(a:b)$-choosable if, for each assignment of lists of $a$ colors to the vertices of $G,$ each vertex can be colored with a set of $b$ colors from its list so that adjacent vertices are colored with disjoint sets. We show that for positive integers $a$ and $b$, every bipartite planar graph is $(a:b)$-choosable iff $\frac{a}{b} \ge 3$. For general planar graphs, we show that if $\frac{a}{b} < 4\frac{2}{5}$, then there exists a planar graph that is not $(a:b)$-choosable, thus improving on a result of X. Zhu, which had $4\frac{2}{9}$. Lastly, we show that every $K_5$-minor-free graph is $(a:b)$-choosable iff $\frac{a}{b} \ge 5$. Along the way, we mention some open problems.

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Glenn G. Chappell. 2022-05-19. List Multicoloring of Planar Graphs and Related Classes. https://arxiv.org/abs/2205.09856

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