2006 | OriginalPaper | Buchkapitel
Thresholds for Path Colorings of Planar Graphs
verfasst von : Glenn G. Chappell, John Gimbel, Chris Hartman
Erschienen in: Topics in Discrete Mathematics
Verlag: Springer Berlin Heidelberg
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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 [Poh90, God91] and, in fact, path 3-choosable [Har97]. 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.