Planar Graphs of Maximum Degree Seven are Class I
نویسندگان
چکیده
منابع مشابه
On total 9-coloring planar graphs of maximum degree seven
Given a graphG, a total k-coloring ofG is a simultaneous coloring of the vertices and edges ofGwith at most k colors. If ∆(G) is themaximum degree ofG, then no graph has a total ∆-coloring, but Vizing conjectured that every graph has a total (∆ + 2)-coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing th...
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It is conjectured by Vizing (1965) that every planar graphs graph G with maximum degree 6 ⩽ ∆ ⩽ 7 is class one. The case ∆ = 7 was confirmed independently by Sanders and Zhao (2001), and by Zhang (2000). In this paper, we prove that every planar graph G with ∆ = 6 and without 7-cycles is class one.
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McDiarmid and Reed [‘On the maximum degree of a random planar graph’, Combin. Probab. Comput. 17 (2008) 591–601] showed that the maximum degree Δn of a random labeled planar graph with n vertices satisfies with high probability (w.h.p.) c1 logn < Δn < c2 logn, for suitable constants 0 < c1 < c2. In this paper, we determine the precise asymptotics, showing in particular that w.h.p. |Δn − c logn|...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 2001
ISSN: 0095-8956
DOI: 10.1006/jctb.2001.2047