The graphs with a symmetrical Euler cycle

TL;DR

This paper characterizes graphs that can be sequenced into symmetrical Euler cycles, focusing on those with automorphisms acting bi-regularly on edges, relevant to boundary cycles in arc-transitive maps.

Contribution

It classifies all graphs with a cyclic automorphism subgroup acting bi-regularly on edges and identifies those with symmetrical Euler cycles.

Findings

Classified graphs with cyclic automorphisms acting bi-regularly on edges.

Identified graphs that can form symmetrical Euler cycles.

Connected these graphs to boundary cycles of arc-transitive maps.

Abstract

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.