Three coloring via triangle counting

TL;DR

This paper advances understanding of 3-colorability in planar graphs by linking triangle face counts to cycle length restrictions, building on prior results and conjectures.

Contribution

It establishes that planar graphs without cycles of lengths 4 to 8 are 3-colorable using a novel combination of counting arguments and existing theorems.

Findings

Planar graphs without cycles of lengths 4-8 are 3-colorable.

Triangles constitute less than 2/3 of faces in certain plane graphs.

The result extends previous partial solutions to Steinberg's conjecture.

Abstract

In the first partial result toward Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly less than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey's resolution of Ore's conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.