Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Congruence classes of orientable 2-cell embeddings of bouquets of circles and dipoles. | Congruence classes of orientable 2-cell embeddings of bouquets of circles and dipoles Yan-Quan Feng Department of Mathematics Beijing Jiaotong University Beijing 100044 . China yqf eng@bj Jin-Ho Kwak Department of Mathematics Pohang University of Science and Technology Pohang 790-784 Korea j inkwak@ Jin-Xin Zhou Department of Mathematics Beijing Jiaotong University Beijing 100044 . China j xzhou@bj Submitted Feb 8 2008 Accepted Mar 1 2010 Published Mar 8 2010 Mathematics Subject Classifications 05C10 05C25 20B25 Abstract Two 2-cell embeddings I X S and J 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 Y of X such that ih YJ. 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 1 Introduction Let X be a finite connected .
