Maps and half-transitive graphs of valency 4 (Q1266391)

A graph is half-transitive if its automorphism group is transitive on vertices and edges but not on arcs of the graph. Among others it is proved that under certain conditions imposed on a regular map, its medial graph is a half-transitive graph of valence 4. Conversely, under certain conditions imposed on the vertex stabilizer, a half-transitive graph of valence 4 gives rise to an irreflexible regular map. This way, new families of half-transitive graphs are constructed from known regular maps.