9 research outputs found
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
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
Fractional coloring of triangle-free planar graphs
We prove that every planar triangle-free graph on vertices has fractional
chromatic number at most
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
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.This is the peer-reviewed version of the following article: dvořák, zdeněk and Bernard Lidický. "3‐Coloring Triangle‐Free Planar Graphs with a Precolored 8‐Cycle." Journal of Graph Theory 80, no. 2 (2015): 98-111, which has been published in final form at doi: 10.1002/jgt.21842. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Posted with permission.</p