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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

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