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引用次数: 0
摘要
托马森（J. Combin. Theory Ser B 128:192-218，2018）证明了每个亚立方平面图都有至多 7 的 2 距离色度数，这最初是由韦格纳猜想的（给定直径的图和着色问题，多特蒙德大学，预印本，1977 年）。在本论文中，我们考虑了这一图形族的三维着色问题，并证明了每一个亚立方平面图形的三维色度数最多为 17，而且我们猜想这个数字可以减少到 12。此外，我们还证明了每一个最大度数至多为 \(\Delta \)的平面图的三维色度数至多为 \((6+o(1))\Delta \)。
Thomassen (J. Combin. Theory Ser B 128:192–218, 2018) showed that every subcubic planar graph has 2-distance chromatic number at most 7, which was originally conjectured by Wegner (graphs with given diameter and a coloring problem, University of Dortmund, preprint, 1977). In this note, we consider 3-distance colorings of this family of graphs, and prove that every subcubic planar graph has 3-distance chromatic number at most 17, and we conjecture that this number can be reduced to 12. In addition, we show that every planar graph with maximum degree at most \(\Delta \) has 3-distance chromatic number at most \((6+o(1))\Delta \).
期刊介绍：
The Bulletin of the Iranian Mathematical Society (BIMS) publishes original research papers as well as survey articles on a variety of hot topics from distinguished mathematicians. Research papers presented comprise of innovative contributions while expository survey articles feature important results that appeal to a broad audience. Articles are expected to address active research topics and are required to cite existing (including recent) relevant literature appropriately. Papers are critically reviewed on the basis of quality in its exposition, brevity, potential applications, motivation, value and originality of the results. The BIMS takes a high standard policy against any type plagiarism. The editorial board is devoted to solicit expert referees for a fast and unbiased review process.
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