Irreducible 4-critical triangle-free toroidal graphs

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 $D>0$ ...

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

Let G be a plane graph of girth at least five. We show that if there exists a 3-coloring φ 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 3coloring extending φ. 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 ...

Edge number critical triangle free graphs with low independence numbers

The structure of all triangle free graphs G = (V,E) with |E|−6|V |+α(G) = 0 is determined, yielding an affirmative answer to a question of Stanis law Radziszowsky and Donald Kreher.

Measuring triangle-free graphs

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

Dvořák, Král’ and Thomas [4, 6] 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.