2021/11/04 by Jiyong Chen, Chen, Jiyong, Cai Heng Li +5
Mathematics · #05C25 #05C35 #20B25 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2111.02615
openalex publication_date 2021/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The edges surrounding a face of a map M form a cycle C, called the boundary cycle of the face, and C is often not a simple cycle. If the map M is arc-transitive, then there is a cyclic subgroup of automorphisms of M which leaves C invariant and is bi-regular on the edges of the induced subgraph [C]; that is to say, C is a symmetrical Euler cycle of [C]. In this paper we determine the family of graphs (which may have multiple edges) whose edge-sets can be sequenced to form a symmetrical Euler cycle. We first classify all graphs which have a cyclic subgroup of automorphisms acting bi-regularly on edges. We then apply this classification to obtain the graphs possessing a symmetrical Euler cycle, and therefore are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arc-transitive maps.