The Semi-Arc Automorphism Group of a Graph with Application to Map Enumeration

A map is a connected topological graph cellularly embedded in a surface. For a given graph , its genus distribution of rooted maps and embeddings on orientable and non-orientable surfaces are separately investigated by many researchers. By introducing the concept of a semi-arc automorphism group of a graph and classifying all its embeddings under the action of its semi-arc automorphism group, we find the relations between its genus distribution of rooted maps and genus distribution of embeddings on orientable and nonorientable surfaces, and give some new formulas for the number of rooted maps on a given orientable surface with underlying graph a bouquet of cycles Bn, a closed-end ladder Ln or a Ringel ladder Rn. A general scheme for enumerating unrooted maps on surfaces(orientable or non-orientable) with a given underlying graph is established. Using this scheme, we obtained the closed formulas for the numbers of non-isomorphic maps on orientable or non-orientable surfaces with an unde...

• Type: Journal
• Year: 2006
• Where: GC
• Authors: Linfan Mao, Yanpei Liu, Erling Wei
• Keywords: Applied Computing, GC 2006, Genus Distribution, Orientable, Semi-arc Automorphism Group