Maps and Half-transitive Graphs of Valency 4

✍ Scribed by D Marušič; R Nedela

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 167 KB
• Volume: 19
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1998.0187

✦ Synopsis

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 correspondence between regular maps and 1 2 -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 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 irreflexible regular map. This way infinite 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 irreflexible 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.