1,015 research outputs found
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
All finite transitive graphs admit self-adjoint free semigroupoid algebras
In this paper we show that every non-cycle finite transitive directed graph
has a Cuntz-Krieger family whose WOT-closed algebra is . This
is accomplished through a new construction that reduces this problem to
in-degree -regular graphs, which is then treated by applying the periodic
Road Coloring Theorem of B\'eal and Perrin. As a consequence we show that
finite disjoint unions of finite transitive directed graphs are exactly those
finite graphs which admit self-adjoint free semigroupoid algebras.Comment: Added missing reference. 16 pages 2 figure
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