List Multicoloring of Planar Graphs and Related Classes
Glenn G. Chappell

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 and , a graph is -choosable if, for each assignment of lists of colors to the vertices of each vertex can be colored with a set of colors from its list so that adjacent vertices are colored with disjoint sets. We show that for positive integers and , every bipartite planar graph is -choosable iff . For general planar graphs, we show that if , then there exists a planar graph that is not -choosable, thus improving on a result of X. Zhu, which had . Lastly, we show that every -minor-free graph is -choosable iff . Along the way, we mention some open problems.
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Taxonomy
TopicsAdvanced Graph Theory Research
