Random 2-cell embeddings of a given graph $G$ are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, $\mathbb{E}[F_G]$, of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar with $n$ nonleaf edges lies in an interval of length $2/(n + 1)$ centered at the expected number of faces of an $n$-edge dipole. This allows us to derive bounds on $\mathbb{E}[F_G]$ for any given graph $G$ in terms of vertex degrees. We conjecture that $\mathbb{E}[F_G ] \le O(n)$ for any simple $n$-vertex graph $G$.

翻译：随机 2 细胞 嵌入给定图形 $G$ 的方法是在每个顶端选择随机的本地旋转 。 我们分析这种嵌入的预期面数, $\ mathbb{E} [F_G]$, 相当于研究其平均基因。 到目前为止, 两个名为单极和双极的家族已知的近似结果。 我们将顶端结果扩展至一个更普通的家族, 称为多星, 即无循环的多图, 其中存在所有边缘的顶点事件 。 具体地说, 我们发现每个有美元非利弗边缘的多星的预期面数在长度为 $/ (n + 1) 的间隔内, 以 $- g$ 的预期面数为中心。 这让我们可以在 $\ mathb{E} [F_ G]$ 上得出任何给定图形的垂直度为$G$. g] 的界限。 我们估计, $\ mathb{ E} [F_ G$_ g$] 。 任何简单的 O (n) g. g.