A proof for Padberg's conjecture on rank of matching polytope

TL;DR

This paper proves Padberg's conjecture that the geometric rank of the matching polytope is exactly one, confirming a key property in the geometric theory of integer polyhedra.

Contribution

We provide a rigorous proof that the geometric rank of the matching polytope is one, settling a longstanding conjecture in polyhedral combinatorics.

Findings

Confirmed Padberg's conjecture on the rank of the matching polytope

Established the geometric rank as exactly one for the matching polytope

Contributed to the understanding of the structure of matching polytopes

Abstract

Padberg introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true.