Further Extensions of the Gr\"{o}tzsch Theorem
Abstract
The Gr\"{o}tzsch Theorem states that every triangle-free planar graph admits a proper -coloring. Among many of its generalizations, the one of Gr\"{u}nbaum and Aksenov, giving -colorability of planar graphs with at most three triangles, is perhaps the most known. A lot of attention was also given to extending -colorings of subgraphs to the whole graph. In this paper, we consider -colorings of planar graphs with at most one triangle. Particularly, we show that precoloring of any two non-adjacent vertices and precoloring of a face of length at most can be extended to a -coloring of the graph. Additionally, we show that for every vertex of degree at most , a precoloring of its neighborhood with the same color extends to a -coloring of the graph. The latter result implies an affirmative answer to a conjecture on adynamic coloring. All the presented results are tight.
Keywords
Cite
@article{arxiv.2110.01862,
title = {Further Extensions of the Gr\"{o}tzsch Theorem},
author = {Hoang La and Borut Lužar and Kenny Štorgel},
journal= {arXiv preprint arXiv:2110.01862},
year = {2022}
}