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Flows on hypermaps

Published online by Cambridge University Press:  18 May 2009

R. Cori  and
A. Machi'
R. Cori
Affiliation:
U. E. R. De Mathématiques et Informatique, Université de Bordeaux I, 351 Cours de la Libération, 33405 Talence Cédex, France
A. Machi'
Affiliation:
Dipartimento di Matematica, Università “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy
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The combinatorial investigation of graphs embedded on surfaces leads one to consider a pair of permutations (σ, α) that generate a transitive group [7]. The permutation α is a fixed-point-free involution and the pair is called a map. When this condition on α is dropped the combinatorial object that arises is called a hypermap. Both maps and hypermaps have a topological description: for maps a classical reference is [13] and for hypermaps such a description can be found in [4] and [6]; a brief account of it will be given below. However, the relationship between maps and hypermaps is not simply that the latter generalize the former. Actually, with every hypermap there is associated a map, its bipartite map, and conversely every bipartite map arises in this way. We do not enter into the details of this question; we refer the reader to the work of Walsh [16]. In this sense hypermaps are, at the same time, a generalization and a special case of maps.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 1 , January 1988 , pp. 17 - 29
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Biggs, N., Algebraic graph theory (Cambridge University Press, 1974).CrossRefGoogle Scholar
2.Brahana, H. R., Systems of circuits on two-dimensional manifolds, Ann. of Math. 23 (1921–1922), 144–168.CrossRefGoogle Scholar
3.Cohn, M. and Lempel, A., Cycle decomposition by disjoint permutations, J. Combinatorial Theory Ser. A 13 (1972), 83–89.CrossRefGoogle Scholar
4.Cori, R., Un code pour les graphes planaires et ses applications, Astérisque No. 27 (Societé Mathematique de France, 1975).Google Scholar
5.Cori, R. and Machi, A., Su alcune proprieta del genere di una coppia di permutazioni, Boll. Un. Mat. Ital. (5) 18A (1981), 84–89.Google Scholar
6.Cori, R., Machì, A., Penaud, J. G. and Vauquelin, B., On the automorphism group of a planar hypermap, European J. Combin. 2 (1981), 331–334.CrossRefGoogle Scholar
7.Edmonds, J., A Combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc. 7, (1960), 646.Google Scholar
8.Jacques, A., Sur le genre d'une paire de substitutions, C. R. Acad. Sci. Paris 267 (1968), 625–627.Google Scholar
9.Jacques, A., Constellations et graphes topologiques, Combinatorial Theory and its applications, Colloq. Math. Soc. Janos Bolyai (1970), 657–673.Google Scholar
10.Jaeger, F., On some algebraic properties of graphs, Progress in graph theory (Waterloo, Oct. 1982) 347–366 (Academic Press, Toronto 1984).Google Scholar
11.Jones, G. A. and Singerman, D., Theory of maps on orientable surfaces, Proc. London Math. Soc. (3) 37 (1978), 273–307.CrossRefGoogle Scholar
12.Machi, A., The Riemann Hurwitz formula for the centralizer of a pair of permutations, Archiv der Math. 42 (1984), 280–288.CrossRefGoogle Scholar
13.Ringel, G., Map colour theorem, (Springer Verlag 1974).CrossRefGoogle Scholar
14.Stahl, S., On the product of certain permutations, to appear.Google Scholar
15.Tutte, W. T., Graph Theory, Encyclopedia of Mathematics and its Applications, 21 (Addison-Wesley, 1984).Google Scholar
16.Walsh, T. R. S., Hypermaps versus bipartite maps, J. Combinatorial Theory Ser. B 18 (1975), 155–163.CrossRefGoogle Scholar