Abstract
A path coloring of a graph $G$ is a vertex coloring of $G$ such that each color class induces a disjoint union of paths. We consider a path-coloring version of list coloring for planar and outerplanar graphs. We show that if each vertex of a planar graph is assigned a list of $3$ colors, then the graph admits a path coloring in which each vertex receives a color from its list. We prove a similar result for outerplanar graphs and lists of size $2$.For outerplanar graphs we prove a multicoloring generalization. We assign each vertex of a graph a list of $q$ colors. We wish to color each vertex with $r$ colors from its list so that, for each color, the set of vertices receiving it induces a disjoint union of paths. We show that we can do this for all outerplanar graphs if and only if $q/r \ge 2$. For planar graphs we conjecture that a similar result holds with $q/r \ge 3$; we present partial results toward this conjecture.
Highlights
All graphs will be finite, simple, and undirected
A graph G is t-defective k-choosable if, whenever each vertex of G is assigned a list of k colors, there exists a vertex coloring of G in which each vertex receives a color from its list, and each color class induces a subgraph of maximum degree at most t
A graph G is (q, r)-choosable, for positive integers q and r, if, whenever each vertex of G is assigned a list of q colors, we may color each vertex with a set of r colors from its list so that adjacent vertices are colored with disjoint sets
Summary
All graphs will be finite, simple, and undirected. See West [27] for graph theoretic terms. A graph G is t-defective k-choosable if, whenever each vertex of G is assigned a list of k colors, there exists a vertex coloring of G in which each vertex receives a color from its list, and each color class induces a subgraph of maximum degree at most t. Gimbel, & Hartman [9, Thm. 3.1] further improved this by showing that each planar graph of girth at least 6 is path 2-choosable. Axenovich, Ueckerdt, & Weiner [5, Thm. 1] improved the Chappell-Gimbel-Hartman result by showing the existence of a path coloring in which each monochromatic component is a path on at most 15 vertices In another improvement of the Skrekovski result, Borodin & Ivanova [6, Thm. 1] showed that each planar graph of girth at least 7 is path 2-choosable such that, in the path coloring, each monochromatic component is a path on at most 3 vertices. For each planar graph G, there exist an infinite number of positive integers m such that G is path (3m, m)-choosable
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