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Abstract.In a previous paper, the authors proved a conjecture of Melnikov that the edges and faces of a plane graph of maximum degree �� may be simultaneously colored with at most ��+3 colors. In this paper, the theorem is reproved with a more direct technique, which also yields improvements. For ��5, the theorem is extended to multigraphs. For ��7, it is shown that ��+2 colors suffice.
Abstract.The problems of cyclic colorings (due to Ore and Plummer) and diagonal colorings (due to Bouchet, Fouquet, Jolivet, and Riviere) were simultaneously generalized by Hornak and Jendrol into d-diagonal colorings. A coloring of a graph embedded on a surface is d-diagonal if any pair of vertices which are in the same face after the deletion of at most d edges of the graph are colored differently...
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