On improving the edge-face coloring theorem

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