P. A. D. S. P. Caldera, S. V. A. Almeida, G. S. Wijesiri
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引用次数: 0
摘要
在拓扑图理论中,图的最大属是一个引人入胜的课题。对于简单连通图 G,最大图属 γM(G)是 G 在其上有 2 单元嵌入的可定向曲面的最大图属。γM(G)有上界,γM(G)≤[β/2],其中β(G)表示贝蒂数,如果相等成立,则称 G 为上可嵌入。在本研究中,通过证明广义彼得森图 GP(n, k) 在 k = 1 和 k = 2 两种情况下的上可嵌入性,确定了在 k = 1 和 k = 2 时 GP(n, k) 的最大属数为 γM(GP(n,k))=[(n+1)/2]。证明的方法是获取生成树 T,并检查 GP(n, k) 的 k = 1 和 k = 2 两种情况下的边补 GP(n, k)\T 中的成分。
The maximum genus of the generalized Petersen Graph, GP (n, k) for the cases k = 1, 2
In Topological graph theory, the maximum genus of graphs has been a fascinating subject. For a simple connected graph G, the maximum genus γM(G) is the largest genus of an orientable surface on which G has a 2-cell embedding. γM(G) has the upper bound, γM(G)≤[β/2], where β(G) denotes the Betti number and G is said to be upper embeddable if the equality holds. In this study, the maximum genus of GP(n, k) is established as γM(GP(n,k))=[(n+1)/2] for k = 1 and k = 2 by proving the upper embeddability of generalized Petersen graph, GP(n, k) for the cases k = 1 and k = 2. The proof is done by obtaining spanning trees T and examining the components in the edge complements GP(n, k)\T for the cases k = 1 and k = 2 of GP(n, k).