Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk
Combinatorics
2017-07-07 v4 Discrete Mathematics
Abstract
Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G, then G has a subgraph H on O(|C|) vertices that also has no 3-coloring extending phi. This is asymptotically best possible and improves a previous bound of Thomassen. In the next paper of the series we will use this result and the attendant theory to prove a generalization to graphs on surfaces with several precolored cycles.
Cite
@article{arxiv.1302.2158,
title = {Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk},
author = {Zdenek Dvorak and Daniel Kral and Robin Thomas},
journal= {arXiv preprint arXiv:1302.2158},
year = {2017}
}
Comments
48 pages, 4 figures This version: Revised according to reviewer comments