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

4-Colorable 6-regular toroidal graphs

AbstractThis paper proves two conjectures of Collins, Fisher and Hutchinson about the chromatic number of some circulant graphs. As a consequence, we characterize 4-colorable 6-regular toroidal graphs

Three-Coloring Graphs Embedded on Surfaces with All Faces Even-Sided

AbstractEvery graph embedded on a surface of positive genus with every face bounded by an even number of edges can be 3-colored provided all noncontractible cycles in the graph are sufficiently long. The bound of three colors is the smallest possible for this type of result