The graphs with a symmetrical Euler cycle
Keywords:edge-transitive graphs, graphs with multiple edges, graph embeddings, arc-transitive maps.
The graphs in this paper are finite, undirected, and without loops, but may have more than one edge between a pair of vertices. If such a graph has ℓ edges, then an Euler cycle is a sequence (e1,e2,…,eℓ) of these ℓ edges, each occurring exactly once, such that ei, ei + 1 are incident with a common vertex for each i (reading subscripts modulo ℓ). An Euler cycle is symmetrical if there exists an automorphism of the graph such that ei → ei + 2 for each i. The cyclic group generated by this automorphism has one orbit on edges if ℓ is odd, or two orbits of length ℓ/2 if ℓ is even: that is to say, the group is regular or bi-regular on edges, respectively. Symmetrical Euler cycles arise naturally from arc-transitive embeddings of graphs in surfaces since, for each face of the embedded graph, the sequence of edges on the boundary of the face forms a symmetrical Euler cycle for the induced subgraph on this edge-set.
We first classify all finite connected graphs which admit a cyclic subgroup of automorphisms that is regular or bi-regular on edges, and identify more than a dozen infinite families of examples. We then prove that exactly six of these families consist of graphs with symmetrical Euler cycles. These are the (only) candidates for the induced subgraphs of the boundary cycles of the faces of arc-transitive maps.