5̃- essential rooted maps on N5 - Wenzhong, Liu; Yanpei, Liu (Utilitas Mathematica, 2007)
A g̃- essential rooted map M on the nonorientable surface N g̃ with genus p is a map embedded on Ng̃ such that M - e is either not a map or a map not embedded on Ng̃ for any edge e of M. In the paper, we first present the explicit formula of counting 5̃- essential rooted maps on Ng̃ with the size as a parameter. Furthermore, the number of singular maps and bouquet maps on N 5̃ are derived.