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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。
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.
期刊介绍:
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.