Third case of the Cyclic Coloring Conjecture

TL;DR

This paper proves the third case of the Cyclic Coloring Conjecture for plane graphs with maximum face size six, extending the known cases and connecting to classical graph coloring theorems.

Contribution

The paper establishes the validity of the Cyclic Coloring Conjecture specifically for D=6, filling a gap in the conjecture's known cases.

Findings

Confirmed the D=6 case of the Cyclic Coloring Conjecture

Extended the understanding of face size constraints in plane graph coloring

Linked the conjecture to existing theorems for D=3 and D=4

Abstract

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 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 D: the case D=3 is equivalent to the Four Color Theorem and the case D=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 D=6 of the conjecture.