Integral bases, perfect matchings, and the Petersen graph

TL;DR

The paper proves that for any matching-covered graph G, the lattice L(G) generated by integral points in the perfect matching polytope P(G) has a lattice basis consisting solely of incidence vectors of perfect matchings. It provides polyhedral proofs of Lovász’s lattice basis result and the CLM results, and a new polyhedral Petersen-graph characterization, avoiding heavy dual-lattice techniques. A key part is the Integral Basis Theorem for Petersen-free graphs, together with the Petersen Graph Lemma and the Intersection Theorem, which control facet structure and intersection properties with separating cuts. The authors develop a composition framework to merge bases across separating cuts, enabling a constructive assembly of integral bases from bricks and near-bricks. The work yields practical polyhedral tools for understanding the lattice structure of P(G) and has implications for the Berge-Fulkerson and 4-flow-type conjectures via lattice decompositions along tight cuts.

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 Lovász (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 $L$. Along the way, we give a new polyhedral characterization of the Petersen graph.

Figures (4)

• Figure 1: The Petersen graph. Any cut separating two $5$-cycles (e.g., the one highlighted) is a separating facet-defining cut.
• Figure 2: Barrier and equivalent cuts. (Left): an example of a near-brick with a tight cut (blue dashed). The corresponding barrier is circled in red. (Right): The two dashed blue cuts are equivalent separating cuts in a near-brick. Contracting two disjoint shores results in $4$-cycle, a bipartite graph.
• Figure 3: Proof of the Intersection Theorem. (Left): Case $1$. the edge $e$ (red dashed) connects two vertices at distance two. The $5$-cycle of $H$ that we could pick is highlighted. (Right): Case $2$. The only maximal barrier is shown by the red box and the options for $e$ are shown as red dashed edges. The $5$-cycle of $H$ that we could pick is highlighted; its choice depends on the perfect matching $M$ (not shown).
• Figure 4: Proof of the Intersection Theorem: Two options for Case $3$. Two maximal barriers are in red and $e$ is shown in dashed. A $5$-cycle of $H$ that we could pick is highlighted; its choice depends on the perfect matching $M$ (not shown). (Left): the contraction vertices of two maximal barriers are adjacent in $H$. (Right): the two contraction vertices are at distance two.

Theorems & Definitions (9)

• Theorem 1.1: CLM3Lovasz87
• Theorem 1.2
• Lemma 1.3
• Theorem 1.4
• Theorem 6.1: ACLS25+
• proof : Proof of \ref{['integral-basis']}
• proof : Proof of \ref{['lattice-basis']}
• proof : Proof of \ref{['composition-lin-ind']}
• proof : Proof of \ref{['composition-lin-comb']}