34 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
Strong edge colorings of graphs and the covers of Kneser graphs
A proper edge coloring of a graph is strong if it creates no bichromatic path
of length three. It is well known that for a strong edge coloring of a
-regular graph at least colors are needed. We show that a -regular
graph admits a strong edge coloring with colors if and only if it covers
the Kneser graph . In particular, a cubic graph is strongly
-edge-colorable whenever it covers the Petersen graph. One of the
implications of this result is that a conjecture about strong edge colorings of
subcubic graphs proposed by Faudree et al. [Ars Combin. 29 B (1990), 205--211]
is false