Locally finite vertex-rotary maps and coset graphs with finite valency and finite edge multiplicity

Abstract

A well-known theorem of Sabidussi shows that a simple G-arc-transitive graph can be represented as a coset graph for the group G. This pivotal result is the standard way to turn problems about simple arc-transitive graphs into questions about groups. In this paper, the Sabidussi representation is extended to arc-transitive, not necessarily simple graphs which satisfy a local-finiteness condition: namely graphs with finite valency and finite edge-multiplicity. The construction yields a G-arc-transitive coset graph $\mathrm{Cos}(G,H,J)$, where $H,J$ are stabilisers in G of a vertex and incident edge, respectively. A first major application is presented concerning arc-transitive maps on surfaces: given a group $G=〈a,z〉$ with $\left|z\right|=2$ and $\left|a\right|$ finite, the coset graph $\mathrm{Cos}(G,〈a〉,〈z〉)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a G-arc-transitive map $(V,E,F)$ (with $V,E,F$ the sets of vertices, edges and faces, respectively), namely, a G-rotary map if $\left|az\right|$ is finite, and a G-bi-rotary map if $\left|z{z}^a\right|$ 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. In addition a coset geometry construction is given of a flag-regular map $(V,E,F)$ for non necessarily simple graphs. For all of these constructions it is proved that the face boundary cycles are simple cycles precisely when the given group acts faithfully on $V\cup F$. Illustrative examples are given for graphs related to the n-dimensional hypercubes and the Petersen graph.