On cyclic colorings and their generalizations Original Research Article

A cyclic coloring is a vertex coloring such that vertices in a face receive different colors. Let Δ be the maximum face degree of a graph. This article shows that plane graphs have cyclic 95Δ-colorings, improving results of Ore and Plummer, and of Borodin. The result is mainly a corollary of a best-possible upper bound on the minimum cyclic degree of a vertex of a plane graph in terms of its maximum face degree. The proof also yields results on the projective plane, as well as for d-diagonal colorings. Also, it is shown that plane graphs with Δ=5 have cyclic 8-colorings. This result and also the 95Δ result are not necessarily best possible.