多重星的随机2-胞腔嵌入
Random 2-cell embeddings of multistars
AI总结：
本文将偶极子图的随机2-胞腔嵌入期望面数结果推广至更一般的多重星族，证明其期望面数区间长度为$2/(n+1)$，并据此推导出任意图基于顶点度数的界限，同时提出简单图期望面数为$O(n)$的猜想。
摘要：
给定图$G$的随机2-胞腔嵌入是通过在每个顶点处选择随机局部旋转得到的。我们分析了此类嵌入的期望面数$\mathbb{E}[F_G]$，这等价于研究其平均亏格。迄今为止，已有针对单极子和偶极子这两类族的紧致结果。我们将偶极子结果推广到称为多重星的更一般族，即存在一个顶点与所有边关联的无环重图。特别地，我们证明了每个具有$n$条非叶子边的多重星的期望面数位于长度为$2/(n + 1)$、以$n$条边偶极子的期望面数为中心的区间内。这使我们能够根据顶点度数推导出任意给定图$G$的$\mathbb{E}[F_G]$界限。我们猜想对于任意简单$n$顶点图$G$，$\mathbb{E}[F_G ] \le O(n)$。
英文摘要：
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 $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
