A Combinatorial Relationship between Eulerian Maps and Hypermaps in Orientable Surfaces

✍ Scribed by D.M. Jackson; T.I. Visentin

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 388 KB
• Volume: 87
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1998.2951

✦ Synopsis

There is a remarkable relationship between the genus series for rooted maps and rooted quadrangulations that has been obtained by a character theoretic argument. Hitherto no combinatorial explanation for this has been given. In this paper we generalize this relationship to a larger set of maps for which such a relationship can be found and we ascertain some of the properties that a putative bijection must possess. Several examples of images of sets under this bijection are given. We give them in detail since they may contain useful information about the bijection.

1999 Academic Press M(u, x, y, z)= : g, i, j, k 0 m g, i, j, k u g x i y j z k for the number m g, i, j, k of rooted maps with i vertices, j faces, k edges and genus g satisfies the following [8] remarkable functional relationship.