Thresholds for Path Colorings of Planar Graphs

Abstract

A graph is path k-colorable if it has a vertex k-coloring in which the subgraph induced by each color class is a disjoint union of paths. A graph is path k-choosable if, whenever each vertex is assigned a list of k colors, such a coloring exists in which each vertex receives a color from its list. It is known that every planar graph is path 3-colorable [15, 13] and, in fact, path 3-choosable [14]. We investigate which planar graphs are path 2-colorable or path 2-choosable. We seek results of a " threshold " nature: on one side of a threshold, every graph is path 2-choosable, and there is a fast coloring algorithm; on the other side, determining even path 2-colorability is NP-complete. We first consider maximum degree. We show that every planar graph with maximum degree at most 4 is path 2-choosable, while for k ≥ 5 it is NP-complete to determine whether a planar graph with maximum degree k is path 2-colorable. Next we consider girth. We show that every planar graph with girth at least 6 is path 2-choosable, while for k ≤ 4 it is NP-complete to determine whether a planar graph with girth k is path 2-colorable. The case of girth 5 remains open.