Classification of ($p,q,n$)-Dipoles on Nonorientable Surfaces

Abstract

A type of rooted map called $(p,q,n)$-dipole, whose numbers on surfaces have some applications in string theory, are defined and the numbers of $(p,q,n)$-dipoles on orientable surfaces of genus 1 and 2 are given by Visentin and Wieler (The Electronic Journal of Combinatorics 14 (2007),#R12). In this paper, we study the classification of $(p,q,n)$-dipoles on nonorientable surfaces and obtain the numbers of $(p,q,n)$-dipoles on the projective plane and Klein bottle.