Arc-transitive cycle decompositions of tetravalent graphs

A cycle decomposition of a graph ▫$Gamma$▫ is a set ▫$mathcal{C}$▫ of cycles of ▫$Gamma$▫ such that every edge of ▫$Gamma$▫ belongs to exactly one cycle in ▫$mathcal{C}$▫. Such a decomposition is called arc-transitive if the group of automorphisms of ▫$Gamma$▫ that preserve setwise acts transitively on the arcs of ▫$Gamma$▫. In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining and enumerating arc-transitive cycle decompositions admitted by a given arc-transitive tetravalent graph. Among other results we show that a connected tetravalent arc-transitive graph is either 2-arc-transitive, or is isomorphic to the medial graph of a reflexible map, or admits exactly one cycle structure