46 research outputs found
Three-coloring triangle-free graphs on surfaces III. Graphs of girth five
We show that the size of a 4-critical graph of girth at least five is bounded
by a linear function of its genus. This strengthens the previous bound on the
size of such graphs given by Thomassen. It also serves as the basic case for
the description of the structure of 4-critical triangle-free graphs embedded in
a fixed surface, presented in a future paper of this series.Comment: 53 pages, 7 figures; updated according to referee remark
Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
We settle a problem of Havel by showing that there exists an absolute
constant d such that if G is a planar graph in which every two distinct
triangles are at distance at least d, then G is 3-colorable. In fact, we prove
a more general theorem. Let G be a planar graph, and let H be a set of
connected subgraphs of G, each of bounded size, such that every two distinct
members of H are at least a specified distance apart and all triangles of G are
contained in \bigcup{H}. We give a sufficient condition for the existence of a
3-coloring phi of G such that for every B\in H, the restriction of phi to B is
constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio
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 phi 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 3-coloring
extending phi. 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 to prove a generalization to graphs on surfaces with
several precolored cycles.Comment: 48 pages, 4 figures This version: Revised according to reviewer
comment
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 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 3-coloring extending. 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 to prove a generalization to graphs on surfaces with several precolored cycles
Three-coloring triangle-free graphs on surfaces I. Extending a coloring to a disk with one triangle
Let G be a plane graph with exactly one triangle T and all other cycles of length at least 5, and let C be a facial cycle of G of length at most six. We prove that a 3-coloring of C does not extend to a 3-coloring of G if and only if C has length exactly six and there is a color x such that either G has an edge joining two vertices of C colored x, or T is disjoint from C and every vertex of T is adjacent to a vertex of C colored x. This is a lemma to be used in a future paper of this series
Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs
Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that for a fixed constant κ, thus generalizing and strengthening several known results. As a corollary, we prove that every triangle-free graph G embedded in a surface of genus g contains a set of vertices such that is 3-colorable
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