977 research outputs found
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
Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains
Aksenov proved that in a planar graph G with at most one triangle, every
precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact
characterization of planar graphs with two triangles in that some precoloring
of a 4-cycle does not extend. We apply this characterization to solve the
precoloring extension problem from two 4-cycles in a triangle-free planar graph
in the case that the precolored 4-cycles are separated by many disjoint
4-cycles. The latter result is used in followup papers to give detailed
information about the structure of 4-critical triangle-free graphs embedded in
a fixed surface.Comment: 38 pages, 6 figures; corrections from the review proces
3-coloring triangle-free planar graphs with a precolored 8-cycle
Let G be a planar triangle-free graph and let C be a cycle in G of length at
most 8. We characterize all situations where a 3-coloring of C does not extend
to a proper 3-coloring of the whole graph.Comment: 20 pages, 5 figure
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 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
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