2022/02/15 by Cai Heng Li, Cheryl E. Praeger, Li, Cai Heng +3 · 5 citations
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2202.07100
openalex publication_date 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 |zza| 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∪ F.