Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle
Discrete Mathematics
2017-03-28 v3
Abstract
Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series.
Cite
@article{arxiv.1010.2472,
title = {Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle},
author = {Zdenek Dvorak and Dan Kral and Robin Thomas},
journal= {arXiv preprint arXiv:1010.2472},
year = {2017}
}
Comments
18 pages, 2 figures; v3: further reviewer remarks incorporated