The graphs with a symmetrical Euler cycle

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 graph has ℓ edges, then an Euler cycle is sequence (e1,e2,…,eℓ) these each occurring exactly once, that ei, ei + 1 incident with common vertex for i (reading subscripts modulo ℓ). An symmetrical if there exists automorphism the → 2 i. cyclic group generated by orbit on edges odd, or two orbits length ℓ/2 even: to say, regular bi-regular respectively. Symmetrical cycles arise naturally from arc-transitive embeddings surfaces since, face embedded graph, boundary forms induced subgraph edge-set. We first classify all finite connected which admit subgroup automorphisms identify dozen infinite families examples. prove six consist cycles. These (only) candidates subgraphs faces maps.