Journal or Book Title
SIAM Journal on Discrete Mathematics
Dvořák, Král', and Thomas [Three-Coloring Triangle-Free Graphs on Surfaces IV. Bounding Face Sizes of 4-Critical Graphs, preprint, arXiv:1404.6356v3, 2015; Three-Coloring Triangle-Free Graphs on Surfaces VI. 3-Colorability of Quadrangulations, preprint, arXiv:1509.01013, 2015] 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 a plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps.
Society for Industrial and Applied Mathematics
Dvorak, Zdenek and Lidicky, Bernard, "Fine structure of 4-critical triangle-free graphs III. General surfaces" (2018). Mathematics Publications. 184.