Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk

TL;DR

This paper proves that in certain plane graphs with girth at least five, a non-extendable 3-coloring of a cycle implies the existence of a small subgraph with the same property, improving previous bounds.

Contribution

It establishes a tight bound on the size of subgraphs responsible for coloring obstructions in plane graphs with girth at least five.

Findings

Existence of small subgraph with no 3-coloring extension

Improved bound over previous results by Thomassen

Foundation for generalization to graphs on surfaces with multiple precolored cycles

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.