Orientable Embedding Distributions By Genus For Certain Type of Non-Planar Graphs (II)

Abstract

In this paper, we give an explicit expression of the genus distributions of $M_j^n$, for $j=1,2,\dots ,11$, which are introduced in the previous paper “Orientable embedding distributions by genus for certain types of non-planar graphs”. For a connected graph $G=(V,E)$ with a cycle, let $e$ be an edge on a cycle. By adding $2n$ vertices $u_1,u_2,u_3\dots ,u_n,v_1,v_2,v_3\dots ,v_n$ on $e$ in sequence and connecting $u_k,v_k$ for $1\le k\le n$, a non-planar graph $G_n$ is obtained for $n\ge 3$. Thus, the orientable embedding distribution of $G_n$ by genus is obtained via the genus distributions of $M_j^n$.