Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs
Combinatorics
2020-08-05 v4
Abstract
Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that the sum of lengths of (>=5)-faces of G is at most linear in g+t+c-1.
Cite
@article{arxiv.1404.6356,
title = {Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs},
author = {Zdenek Dvorak and Daniel Kral and Robin Thomas},
journal= {arXiv preprint arXiv:1404.6356},
year = {2020}
}
Comments
28 pages, 1 figure v2: Several changes necessitated by further papers in the series included. Fixed an error: excluding only separating non-contractible 4-cycles is not enough. v3: Added a result on number of vertices needed to remove to make a graph 3-colorable v4: Updated for reviewer comments