On the cyclic coloring conjecture

Abstract

A cyclic coloring of a plane graph $G$ is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph $G$ is its cyclic chromatic number $\chi_c(G)$. Let $\Delta^*(G)$ be the maximum face degree of a graph $G$. In this note we show that to prove the Cyclic Coloring Conjecture of Borodin from 1984, saying that every connected plane graph $G$ has $\chi_c(G) \leq \lfloor \frac{3}{2}\Delta^*(G)\rfloor$, it is enough to do it for subdivisions of simple $3$-connected plane graphs. We have discovered four new different upper bounds on $\chi_c(G)$ for graphs $G$ from this restricted family; three bounds of them are tight. As corollaries, we have shown that the conjecture holds for subdivisions of plane triangulations, simple $3$-connected plane quadrangulations, and simple $3$-connected plane pentagulations with an even maximum face degree, for regular subdivisions of simple $3$-connected plane graphs of maximum degree at least 10, and for subdivisions of simple $3$-connected plane graphs having the maximum face degree large enough in comparison with the number of vertices of their longest paths consisting only of vertices of degree two.