Maps, One-regular Graphs and Half-transitive Graphs of Valency 4

A subgroup G of automorphisms of a graph X is said to be 1 2-transitive if it is vertex and edge but not arc-transitive. The graph X is said to be 1 2-transitive if Aut X is 1 2-transitive. The graph X is called one-regular if Aut X acts regularly on the set arcs of X. The interplay of three diierent concepts of maps, one-regular graphs and 1 2-transitive group actions on graphs of valency 4 is investigated. The correspondence between regular maps and 1 2-transitive group actions on graphs of valency 4 is given via the well known concept of medial graphs. Among others it is proved that under certain general conditions imposed on a map, its medial graph must be a 1 2-transitive graph of valency 4, and vice-versa, under certain conditions imposed on the vertex stabilizer, a 1 2-transitive graph of valency 4 gives rise to an irreeexible regular map. This way innnite families of 1 2-transitive graphs are constructed from known examples of regular maps. Conversely , known constructions of 1 2-transitive graphs of valency 4 give rise to new examples of irreeexible regular maps. In the end, the concept of a symmetric genus of a 1 2-transitive graph of valency 4 is introduced. In particular, 1 2-transitive graphs of valency 4 and small symmetric genuses are discussed.