A Note on Coloring Vertex-transitive Graphs

We prove bounds on the chromatic number χ of a vertex-transitive graph in terms of its clique number ω and maximum degree ∆. We conjecture that every vertex-transitive graph satisfies χ 6 max{ω, ⌈ 5∆+3 6 ⌉ }, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with ∆ > 13 we prove the Borodin–Kostochka conjecture, i.e., χ 6 max{ω,∆− 1}.