Locally Finite Vertex-Rotary Maps and Coset Graphs with Finite Valency and Finite Edge Multiplicity

arXiv2202.07100MaRDI QIDQ6391122

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Publication date: 14 February 2022

Abstract: 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 $G=la,z$ with $|z|=2$ and $|a|$ finite, the coset graph $Cos(G,la,lz)$ 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 $|z{z}^a|$ 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 $VcupF$.

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