On Improving the Edge-Face Coloring Theorem

Abstract.

 In a previous paper, the authors proved a conjecture of Melnikov that the edges and faces of a plane graph of maximum degree Δ may be simultaneously colored with at most Δ+3 colors. In this paper, the theorem is reproved with a more direct technique, which also yields improvements. For Δ≤5, the theorem is extended to multigraphs. For Δ≥7, it is shown that Δ+2 colors suffice.