In 1965, Vizing proved that planar graphs of maximum degree at least eight have the edge chromatic number equal to their maximum degree. He conjectured the same if the maximum degree is either six or seven. This article proves the maximum degree seven case.2001 Elsevier Science

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Abstract.It is shown that a planar graph without/-circuits, 4 < i < 9, is 3-colorable. This result strengthens the result obtained

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