  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.
