Congruence Classes of Orientable 2-Cell Embeddings of Bouquets of Circles and Dipoles

Two 2-cell embeddings ı : X → S and  : X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h : S → S and a graph automorphism γ of X such that ıh = γ. Mull et al. [Proc. Amer. Math. Soc. 103(1988) 321–330] developed an approach for enumerating the congruence classes of 2-cell embeddings of a simple graph (without loops and multiple edges) into closed orientable surfaces and as an application, two formulae of such enumeration were given for complete graphs and wheel graphs. The approach was further developed by Mull [J. Graph Theory 30(1999) 77–90] to obtain a formula for enumerating the congruence classes of 2cell embeddings of complete bipartite graphs into closed orientable surfaces. By considering automorphisms of a graph as permutations on its dart set, in this paper Mull et al.’s approach is generalized to any graph with loops or multiple edges, and by using this method we enumerate the congruence classes of 2-cell embeddings of a bouquet of circles and a dipole into closed orientable surfaces. ∗This work was supported by the National Natural Science Foundation of China (10871021,10901015), the Specialized Research Fund for the Doctoral Program of Higher Education in China (20060004026), and Korea Research Foundation Grant (KRF-2007-313-C00011) in Korea. the electronic journal of combinatorics 17 (2010), #R41 1