Cyclic Coloring of Plane Graphs with Maximum Face Size 16 and 17

TL;DR

This paper proves a longstanding conjecture that 3-connected plane graphs with maximum face sizes 16 and 17 can be vertex-colored with at most D+2 colors without face conflicts, extending previous results.

Contribution

The authors confirm the Plummer and Toft conjecture for maximum face sizes 16 and 17, filling gaps in the known cases.

Findings

Conjecture holds for D=16 and D=17.

Extends the class of plane graphs with proven cyclic coloring bounds.

Supports the conjecture's validity for all D≥3.

Abstract

Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size D can be colored using at most D+2 colors in such a way that no face is incident with two vertices of the same color. The conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for D=16 and D=17.