We prove bounds on the chromatic number Ο of a vertex-transitive graph
in terms of its clique number Ο and maximum degree Ξ. We
conjecture that every vertex-transitive graph satisfies Οβ€max{Ο,β65Ξ+3ββ} and we
prove results supporting this conjecture. Finally, for vertex-transitive graphs
with Ξβ₯13 we prove the Borodin-Kostochka conjecture, i.e.,
Οβ€max{Ο,Ξβ1}