外-1-平面图的入射着色

Pub Date : 2024-06-05 DOI:10.1007/s10255-024-1126-3

Meng-ke Qi, Xin Zhang

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引用次数: 0

摘要

如果一个图形可以在平面上绘制，使得所有顶点都位于外面上，并且每条边最多与另一条边交叉，那么这个图形就是外-1-平面图形。众所周知，每个外-1-平面图都是一个平面局部 3 树。在本文中，我们猜想每个平面图 G 都有一个适当的入射 (Δ(G) + 2)- 着色，并针对最大度数至少为 8 或周长至少为 4 的外-1 平面图证实了这一猜想。具体地说，我们证明了每一个外-1-平面图 G 都有一个入射 (Δ(G) + 3, 2)-着色，而每一个最大度数至少为 8 或周长至少为 4 的外-1-平面图 G 都有一个入射 (Δ(G) + 2, 2)-着色。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Incidence Coloring of Outer-1-planar Graphs

A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge. It is known that every outer-1-planar graph is a planar partial 3-tree. In this paper, we conjecture that every planar graph G has a proper incidence (Δ(G) + 2)-coloring and confirm it for outer-1-planar graphs with maximum degree at least 8 or with girth at least 4. Specifically, we prove that every outer-1-planar graph G has an incidence (Δ(G) + 3, 2)-coloring, and every outer-1-planar graph G with maximum degree at least 8 or with girth at least 4 has an incidence (Δ(G) + 2, 2)-coloring.

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