标题： 积分基，完美匹配和彼得森图 显示英文标题

标题： Integral bases, perfect matchings, and the Petersen graph

摘要： 设 $G=(V,E)$ 为一个匹配覆盖图，记 $P$ 为其完美匹配多面体，记 $L$ 为由 $P$ 中的整数点生成的整数格。在本文中，我们给出了对 Lovász（1987）建立的两个困难结果以及 Carvalho、Lucchesi 和 Murty（2002）在三篇系列论文中建立的结果的多面体证明。 更具体地说，我们重新证明了对于所有$x\in \mathrm{lin}(P)\cap \mathbb{Z}^E$，$L$的格基仅由$G$和$2x\in L$的某些完美匹配的关联向量组成，如果$G$没有彼得森砖块，则$L = \mathrm{lin}(P)\cap \mathbb{Z}^E$。 这是通过研究$P$的面部结构及其与格子$L$的关系来实现的。在此过程中，我们给出了彼得森图的新多面体特征。 显示英文摘要

摘要： Let $G=(V,E)$ be a matching-covered graph, denote by $P$ its perfect matching polytope, and by $L$ the integer lattice generated by the integral points in $P$. In this paper, we give polyhedral proofs for two difficult results established by Lov\'{a}sz (1987), and by Carvalho, Lucchesi, and Murty (2002) in a series of three papers. More specifically, we reprove that $L$ has a lattice basis consisting solely of incidence vectors of some perfect matchings of $G$, $2x\in L$ for all $x\in \mathrm{lin}(P)\cap \mathbb{Z}^E$, and if $G$ has no Petersen brick then $L = \mathrm{lin}(P)\cap \mathbb{Z}^E$. This is achieved by studying the facial structure of $P$ and its relationship with the lattice $L$. Along the way, we give a new polyhedral characterization of the Petersen graph.