Maps and Half-transitive Graphs of Valency 4

Abstract

A subgroupG of automorphisms of a graphX is said to be 12- transitive if it is vertex- and edge- but not arc-transitive. The graphX is said to be 12-transitive if AutX is 12-transitive. The correspondence between regular maps and 12-transitive group actions on graphs of valency 4 is studied 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 12-transitive graph of valency 4 and, vice versa, under certain conditions imposed on the vertex stabilizer, a 12-transitive graph of valency 4 gives rise to\\break an irreflexible regular map. This way infinite families of 12-transitive graphs are constructed from known examples of regular maps. Conversely, known constructions of 12-transitive graphs of valency 4 give rise to new examples of irreflexible regular maps. In the end, the concept of a symmetric genus of a 12-transitive graph of valency 4 is introduced. In particular, 12-transitive graphs of valency 4 and small symmetric genuses are discussed.