In 1965, Vizing proved that planar graphs of maximum degree at least eight have the edge chromati... more 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.
... On total 9-coloring planar graphs of maximum degree seven. Daniel P. Sanders 1 ,; Yue Zhao 2.... more ... On total 9-coloring planar graphs of maximum degree seven. Daniel P. Sanders 1 ,; Yue Zhao 2. Article first published online: 13 MAY 1999. ... Email: Daniel P. Sanders ([email protected]) Yue Zhao ([email protected]). Publication History. ...
In 1969, Ore and Plummer defined an angular coloring as a natural extension of the Four Color Pro... more In 1969, Ore and Plummer defined an angular coloring as a natural extension of the Four Color Problem: a face coloring of a plane graph where faces meeting even at a vertex must have distinct colors. A natural lower bound is the maximum degree Δ of the graph. Some graphs require ⌊32Δ⌋ colors in an angular coloring. Ore and Plummer gave an upper bound of 2Δ, which was improved to ⌊95Δ⌋ by the authors with Borodin. This article gives a new upper bound of ⌈59Δ⌉ on the angular chromatic number. The cyclic chromatic number is the equivalent dual vertex coloring problem.
A cyclic coloring is a vertex coloring such that vertices in a face receive di erent colors. Let ... more A cyclic coloring is a vertex coloring such that vertices in a face receive di erent colors. Let be the maximum face degree of a graph. This article shows that plane graphs have cyclic 9 5 -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 = 5 have cyclic 8-colorings. This result and also the 9 5 result are not necessarily best possible.
It is shown that a planar graph withouti-circuits, 4 ≤i ≤ 9, is 3-colorable. This result strength... more It is shown that a planar graph withouti-circuits, 4 ≤i ≤ 9, is 3-colorable. This result strengthens the result obtained by H.L. Abbott and B. Zhou.
In 1965, Vizing proved that planar graphs of maximum degree at least eight have the edge chromati... more 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.
... On total 9-coloring planar graphs of maximum degree seven. Daniel P. Sanders 1 ,; Yue Zhao 2.... more ... On total 9-coloring planar graphs of maximum degree seven. Daniel P. Sanders 1 ,; Yue Zhao 2. Article first published online: 13 MAY 1999. ... Email: Daniel P. Sanders ([email protected]) Yue Zhao ([email protected]). Publication History. ...
In 1969, Ore and Plummer defined an angular coloring as a natural extension of the Four Color Pro... more In 1969, Ore and Plummer defined an angular coloring as a natural extension of the Four Color Problem: a face coloring of a plane graph where faces meeting even at a vertex must have distinct colors. A natural lower bound is the maximum degree Δ of the graph. Some graphs require ⌊32Δ⌋ colors in an angular coloring. Ore and Plummer gave an upper bound of 2Δ, which was improved to ⌊95Δ⌋ by the authors with Borodin. This article gives a new upper bound of ⌈59Δ⌉ on the angular chromatic number. The cyclic chromatic number is the equivalent dual vertex coloring problem.
A cyclic coloring is a vertex coloring such that vertices in a face receive di erent colors. Let ... more A cyclic coloring is a vertex coloring such that vertices in a face receive di erent colors. Let be the maximum face degree of a graph. This article shows that plane graphs have cyclic 9 5 -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 = 5 have cyclic 8-colorings. This result and also the 9 5 result are not necessarily best possible.
It is shown that a planar graph withouti-circuits, 4 ≤i ≤ 9, is 3-colorable. This result strength... more It is shown that a planar graph withouti-circuits, 4 ≤i ≤ 9, is 3-colorable. This result strengthens the result obtained by H.L. Abbott and B. Zhou.
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