Abstract
A map is a connected topological graph cellularly embedded in a surface. For a given graph Γ, 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 non-orientable surfaces, and give some new formulas for the number of rooted maps on a given orientable surface with underlying graph a bouquet of cycles B n , a closed-end ladder L n or a Ringel ladder R n . 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 underlying bouquet B n in this paper.
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Mao, L., Liu, Y. & Wei, E. The Semi-Arc Automorphism Group of a Graph with Application to Map Enumeration. Graphs and Combinatorics 22, 83–101 (2006). https://doi.org/10.1007/s00373-006-0637-4
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DOI: https://doi.org/10.1007/s00373-006-0637-4