1,132 research outputs found
Further Extensions of the Gr\"{o}tzsch Theorem
The Gr\"{o}tzsch Theorem states that every triangle-free planar graph admits
a proper -coloring. Among many of its generalizations, the one of
Gr\"{u}nbaum and Aksenov, giving -colorability of planar graphs with at most
three triangles, is perhaps the most known. A lot of attention was also given
to extending -colorings of subgraphs to the whole graph. In this paper, we
consider -colorings of planar graphs with at most one triangle.
Particularly, we show that precoloring of any two non-adjacent vertices and
precoloring of a face of length at most can be extended to a -coloring
of the graph. Additionally, we show that for every vertex of degree at most
, a precoloring of its neighborhood with the same color extends to a
-coloring of the graph. The latter result implies an affirmative answer to a
conjecture on adynamic coloring. All the presented results are tight
Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
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
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