List Multicoloring of Planar Graphs and Related Classes

TL;DR

This paper characterizes the conditions under which various classes of planar and minor-free graphs are list multicolorable, establishing thresholds for the ratio of colors needed for choosability.

Contribution

It provides exact thresholds for $(a:b)$-choosability of bipartite planar, general planar, and $K_5$-minor-free graphs, improving previous bounds and resolving specific cases.

Findings

Bipartite planar graphs are $(a:b)$-choosable iff a/b ≥ 3.

For general planar graphs, if a/b < 4 2/5, some are not $(a:b)$-choosable.

Every $K_5$-minor-free graph is $(a:b)$-choosable iff a/b ≥ 5.

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.