5̃- essential rooted maps on N5

Abstract

A g̃- essential rooted map M on the nonorientable surface N _g̃ with genus p is a map embedded on N_g̃ such that M - e is either not a map or a map not embedded on N_g̃ for any edge e of M. In the paper, we first present the explicit formula of counting 5̃- essential rooted maps on N_g̃ with the size as a parameter. Furthermore, the number of singular maps and bouquet maps on N _5̃ are derived.