#### Campus Units

Mathematics

#### Document Type

Article

#### Publication Version

Published Version

#### Publication Date

2017

#### Journal or Book Title

SIAM Journal on Discrete Mathematics

#### Volume

31

#### Issue

2

#### First Page

865

#### Last Page

874

#### DOI

10.1137/15M1023397

#### 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 if $S\subseteq V(H)$ 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.

#### Copyright Owner

The Authors

#### Copyright Date

2017

#### Language

en

#### File Format

application/pdf

#### Recommended Citation

Dvořák, Zdeněk and Lidicky, Bernard, "Fine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-Cycles" (2017). *Mathematics Publications*. 120.

https://lib.dr.iastate.edu/math_pubs/120

## Comments

This article is published as Dvorák, Zdenek, and Bernard Lidický. "Fine structure of 4-critical triangle-free graphs II. Planar triangle-free graphs with two precolored 4-cycles."

SIAM Journal on Discrete Mathematics31, no. 2 (2017): 865-874. doi: 10.1137/15M1023397. Posted with permission.