局部有限点旋转映射与有限度有限边重陪集图
Locally Finite Vertex-Rotary Maps and Coset Graphs with Finite Valency and Finite Edge Multiplicity
摘要：
一个简单的$G$ -弧传递图可以表示为群$G$的陪集图。将这种表示推广到具有有限价和有限边重数的$G$ -弧传递陪集图$\Cos(G,H,J)$的构造，其中$H, J$分别是$G$中顶点和入射边的稳定子.给定一个群$G=ła,z\r$，其$|z|=2$和$|a|$是有限的，在适当的有限性假设下，陪集图$\Cos(G,ła\r,łz\r)$具有两个不同的弧传递嵌入，即$G$ -弧传递映射$(V,E,F)$，如果$|az|$是有限的，则具有一个{ \it $G$ -rotary}映射，如果$|zz^a|$是有限的，则具有一个{ \it $G$ -bi-rotary}映射. $G$ -旋转映射可以表示为$G$的陪集几何，扩展了陪集图的概念。然而，$G$ -双向旋转贴图没有这样的表示，并且除了面和边之间的关联之外，还必须指定面边界循环。给出了旗正则映射$(V,E,F)$的陪集几何构造。在所有这些构造中，我们证明了当给定群忠实地作用在$V\cup F$上时，面边界圈是正则圈，而正则圈是简单圈。
英文摘要：
It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite edge-multiplicity, where $H, J$ are stabilisers in $G$ of a vertex and incident edge, respectively. Given a group $G=ła,z\r$ with $|z|=2$ and $|a|$ finite, the coset graph $\Cos(G,ła\r,łz\r)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a $G$-arc-transitive map $(V,E,F)$, namely, a {\it $G$-rotary} map if $|az|$ is finite, and a {\it $G$-bi-rotary} map if $|zz^a|$ is finite. The $G$-rotary map can be represented as a coset geometry for $G$, extending the notion of a coset graph. However the $G$-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. We also give a coset geometry construction of a flag-regular map $(V,E,F)$. In all of these constructions we prove that the face boundary cycles are regular cycles which are simple cycles precisely when the given group acts faithfully on $V\cup F$.
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