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  • 标题:A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable
  • 本地全文:下载
  • 作者:Enqiang Zhu ; Yongsheng Rao
  • 期刊名称:Discrete Dynamics in Nature and Society
  • 印刷版ISSN:1026-0226
  • 电子版ISSN:1607-887X
  • 出版年度:2020
  • 卷号:2020
  • 页码:1-8
  • DOI:10.1155/2020/3196540
  • 出版社:Hindawi Publishing Corporation
  • 摘要:A total k -coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total Δ G + 2 -coloring, where Δ G is the maximum degree of G . This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of Δ G ≥ 9 or Δ G ∈ 7,8 with some restrictions has a total Δ G + 1 -coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p , q , ℓ for some p , q , ℓ ∈ 3,4,4 , 3,3,4 .
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