Table of Contents
ISRN Discrete Mathematics
Volume 2013, Article ID 712431, 12 pages
http://dx.doi.org/10.1155/2013/712431
Research Article

A Bijection for Tricellular Maps

Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark

Received 30 July 2013; Accepted 27 September 2013

Academic Editors: A. Kelarev and W. F. Smyth

Copyright © 2013 Hillary S. W. Han and Christian M. Reidys. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We give a bijective proof for a relation between unicellular, bicellular, and tricellular maps. These maps represent cell complexes of orientable surfaces having one, two, or three boundary components. The relation can formally be obtained using matrix theory (Dyson, 1949) employing the Schwinger-Dyson equation (Schwinger, 1951). In this paper we present a bijective proof of the corresponding coefficient equation. Our result is a bijection that transforms a unicellular map of genus into unicellular, bicellular or tricellular maps of strictly lower genera. The bijection employs edge cutting, edge contraction, and edge deletion.