Integral bases, perfect matchings, and the Petersen graph

Abstract

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 short, 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 totaling over 120 pages. More specifically, we prove 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 $L = \mathrm{lin}(P)\cap \mathbb{Z}^E$. Our proof avoids major technical aspects of the previous proofs, the most important of these being a characterization of the dual lattice, and a `Petersen-brick-sensitive' ear decomposition result for matching-covered graphs. This is achieved by a novel study of the facial structure of the polytope $P$ and its relationship with the lattice $L$. It is also based on a first-of-its-kind polyhedral characterization of the Petersen graph.