Three coloring via triangle counting
Combinatorics
2022-09-13 v3
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.
Cite
@article{arxiv.2203.08136,
title = {Three coloring via triangle counting},
author = {Zachary Hamaker and Vincent Vatter},
journal= {arXiv preprint arXiv:2203.08136},
year = {2022}
}