Fine structure of 4-critical triangle-free graphs II. Planar triangle-free graphs with two precolored 4-cycles

TL;DR

This paper investigates 3-coloring extensions in triangle-free planar graphs with two distant precolored 4-cycles, identifying structural conditions that determine colorability and confirming a conjecture related to coloring extensions.

Contribution

It establishes conditions under which precolorings extend in triangle-free planar graphs and proves a conjecture about coloring extensions in graphs with distant precolored vertices.

Findings

Precoloring extension depends on specific substructures in the graph.

Existence of a constant D ensuring extension for vertices at large distances.

Provides exponential lower bounds on the number of 3-colorings.

Abstract

We study 3-coloring properties of triangle-free planar graphs $G$ with two precolored 4-cycles $C_1$ and $C_2$ that are far apart. We prove that either every precoloring of $C_1\cup C_2$ extends to a 3-coloring of $G$, or $G$ contains one of two special substructures which uniquely determine which 3-colorings of $C_1\cup C_2$ extend. As a corollary, we prove that there exists a constant $D>0$ such that if $H$ is a planar triangle-free graph and $S\subseteq V(H)$ consists of vertices at pairwise distances at least $D$, then every precoloring of $S$ extends to a 3-coloring of $H$. This gives a positive answer to a conjecture of Dvo\v{r}\'ak, Kr\'al' and Thomas, and implies an exponential lower bound on the number of 3-colorings of triangle-free planar graphs of bounded maximum degree.