2,815 research outputs found
A note on coloring vertex-transitive graphs
We prove bounds on the chromatic number of a vertex-transitive graph
in terms of its clique number and maximum degree . We
conjecture that every vertex-transitive graph satisfies and we
prove results supporting this conjecture. Finally, for vertex-transitive graphs
with we prove the Borodin-Kostochka conjecture, i.e.,
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
Every planar graph with the Liouville property is amenable
We introduce a strengthening of the notion of transience for planar maps in
order to relax the standard condition of bounded degree appearing in various
results, in particular, the existence of Dirichlet harmonic functions proved by
Benjamini and Schramm. As a corollary we obtain that every planar non-amenable
graph admits Dirichlet harmonic functions
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