Congruence Classes of Orientable 2-Cell Embeddings of Bouquets of Circles and Dipoles
Congruence Classes of Orientable 2-Cell Embeddings of Bouquets of Circles and Dipoles
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DOI：
10.37236/313
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发表时间：
2010-03
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影响因子：
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通讯作者：
Yan-Quan Feng;J. Kwak;Jin-Xin Zhou
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作者：
Yan-Quan Feng;J. Kwak;Jin-Xin Zhou
如果存在方向保持表面同态 $h : S \to S$ 和 $X$ 的图自同构 $\gamma$ 使得 $\imath h =\gamma\jmath$，则两个 $2$ 单元嵌入 $\imath : X \to S$ 和 $\jmath : X \to S$ 到闭合可定向表面 $S$ 中是全等的。穆尔等人。 [程序。阿米尔。数学。苏克。 103(1988) 321–330]开发了一种将简单图（没有循环和多边）的 $2$-cell 嵌入枚举到闭合可定向表面的同余类的方法，并且作为一个应用，给出了完整图和轮图的两个枚举公式。 Mull 进一步发展了该方法 [J.图论 30(1999) 77–90] 获得一个公式，用于枚举完整二部图的 $2$-cell 嵌入到闭合可定向曲面中的同余类。通过将图的自同构视为其飞镖集上的排列，在本文中，Mull 等人的方法被推广到任何具有环或多边的图，并且通过使用此方法，我们枚举了将一束圆和偶极子嵌入到闭合可定向曲面中的 $2$ 单元的同余类。
Two $2$-cell embeddings $\imath : X \to S$ and $\jmath : X \to S$ of a connected graph $X$ into a closed orientable surface $S$ are congruent if there are an orientation-preserving surface homeomorphism $h : S \to S$ and a graph automorphism $\gamma$ of $X$ such that $\imath h =\gamma\jmath$. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of $2$-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77–90] to obtain a formula for enumerating the congruence classes of $2$-cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.'s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of $2$-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces.