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August 1, 2026Mathematical ProgrammingOpen Access

Integral bases, perfect matchings, and the Petersen graph

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Authors

AAAhmad AbdiLondon School of Economics and Political ScienceOSOlha SilinaCarnegie Mellon University

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Implication

Demonstrates lattice properties of perfect matchings in matching-covered graphs, indicating a new approach to previous complex proofs.

Key Points

  • The research aims to simplify and provide new insights into the lattice structure of perfect matching polytopes in matching-covered graphs.
  • Provided polyhedral proofs for results concerning perfect matching polytopes and their integer lattices.
  • Examined the facial structure of the perfect matching polytope and its relationship with the integer lattice.
  • Developed a novel characterization of the Petersen graph.
  • Demonstrated that the integer lattice has a basis formed entirely by incidence vectors of perfect matchings.
  • Showed that for any integral point x in the polytope, 2x belongs to the integer lattice.
  • Established that if the graph has no Petersen brick, the integer lattice equals the linear space intersection with the integers.

Cite This Study

Abdi et al. (2026) studied this question.

synapsesocial.com/papers/6a6d97c4e258b358b3c6aacfhttps://doi.org/10.1007/s10107-026-02401-w
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1Strongly Connected Orientations and Integer Lattices2026
  2. 2Matching polytopes, Gorensteinness, and the integer decomposition property2024
  3. 3The Λ-Spectrum of Cubic Graphs: Perfect Matching Exchange and the Erdos-Gyárfás (falsified)2026
  4. 4Perfect matchings and spanning trees: squarishness, bijections and independence2024
  5. 5One-ended spanning trees and definable combinatorics2024 · 3 citations