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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.,
4-Factor-criticality of vertex-transitive graphs
A graph of order is -factor-critical, where is an integer of the
same parity as , if the removal of any set of vertices results in a
graph with a perfect matching. 1-factor-critical graphs and 2-factor-critical
graphs are well-known factor-critical graphs and bicritical graphs,
respectively. It is known that if a connected vertex-transitive graph has odd
order, then it is factor-critical, otherwise it is elementary bipartite or
bicritical. In this paper, we show that a connected vertex-transitive
non-bipartite graph of even order at least 6 is 4-factor-critical if and only
if its degree is at least 5. This result implies that each connected
non-bipartite Cayley graphs of even order and degree at least 5 is
2-extendable.Comment: 34 pages, 3 figure
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