Three coloring via triangle counting [article]

Zachary Hamaker, Vincent Vatter
2022 arXiv   pre-print
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
more » ... r k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.
arXiv:2203.08136v3 fatcat:t23dxzkpnvdynnl5cqcj4og2gq