Cyclic Colorings of 3-Polytopes with Large Maximum Face Size

Abstract

A cyclic coloring of a plane graph is a vertex coloring in which, for each face f, all the vertices in the boundary of f have different colors. It is proved that if G is a 3-polytope (3-connected plane graph) with maximum face size ${\mathnormal{\Delta}^*}$, then G is cyclically $(\mathnormal{\Delta}^* + 1)$-colorable provided that $\mathnormal{\Delta}^*$ is large enough. The figure $\mathnormal{\Delta}^* + 1$ is sharp.