Fine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-Cycles

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 $D\u3e0$ 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\u27, and Thomas, and implies an exponential lower bound on the number of 3-colorings of triangle-free planar graphs of bounded maximum degree

Fine structure of 4-critical triangle-free graphs III. General surfaces

Dvo\v{r}\'ak, Kr\'al' and Thomas gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description however contains two substructures (both related to graphs embedded in plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps.Comment: 15 pages, 1 figure; corrections from the review process. arXiv admin note: text overlap with arXiv:1509.0101

Fine Structure of 4-Critical Triangle-Free Graphs II. Planar Triangle-Free Graphs with Two Precolored 4-Cycles

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.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 Mathematics 31, no. 2 (2017): 865-874. doi: 10.1137/15M1023397. Posted with permission.</p