Third Case of the Cyclic Coloring Conjecture

Abstract

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size $\Delta^*$ can be colored using at most $\lfloor3\Delta^*/2\rfloor$ colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring Conjecture has been proven only for two values of $\Delta^*$: the case $\Delta^*=3$ is equivalent to the Four Color Theorem and the case $\Delta^*=4$ is equivalent to Borodin's Six Color Theorem, which says that every graph that can be drawn in the plane with each edge crossed by at most one other edge is 6-colorable. We prove the case $\Delta^*=6$ of the conjecture.