On the cyclic coloring conjecture

TL;DR

This paper reduces the proof of the cyclic coloring conjecture to subdivisions of simple 3-connected plane graphs and introduces new upper bounds on the cyclic chromatic number for these graphs, confirming the conjecture in several cases.

Contribution

It shows that proving the conjecture for all plane graphs can be limited to subdivisions of simple 3-connected plane graphs and provides new tight bounds for these cases.

Findings

Four new upper bounds on the cyclic chromatic number for subdivisions of simple 3-connected plane graphs.

The cyclic coloring conjecture holds for subdivisions of plane triangulations, quadrangulations, and pentagulations under certain conditions.

The conjecture is verified for regular subdivisions of 3-connected plane graphs with high maximum degree.

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.