The graphs with a symmetrical Euler cycle

Dedicated to our friend and colleague Marston Conder on the occasion of his 65th birthday. 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-set can be sequenced to form a symmetrical Euler cycle. We first classify all graphs and 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 subgraph of the boundary cycle of a face in an arc-transitive map.