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

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 Dvořák, Král' and Thomas, and implies an exponential lower bound on the number of 3-colorings of triangle-free planar graphs of bounded maximum degree.

12 pages, 2 figures

References in corpus (2)

• Planar 4-critical graphs with four triangles
• 4-critical graphs on surfaces without contractible (<=4)-cycles

Cited by in corpus (3)

• Flexibility of triangle-free planar graphs
• Fine structure of 4-critical triangle-free graphs III. General surfaces
• Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains