On 2-partitionable clutters and the MFMC property

TL;DR

This paper introduces 2-partitionable clutters, explores their properties, and provides an infinite family verifying the Conforti-Cornuéjols conjecture, linking combinatorial and algebraic aspects of clutters.

Contribution

It defines 2-partitionable clutters, studies their properties, and identifies a new infinite family that satisfies the conjecture, connecting combinatorics and algebra.

Findings

Found a new infinite family of 2-partitionable clutters verifying the conjecture.

Analyzed properties of the incidence matrix and minors of these clutters.

Explored the algebraic normality of the associated Rees algebra.

Abstract

We introduce 2-partitionable clutters as the simplest case of the class of $k$-partitionable clutters and study some of their combinatorial properties. In particular, we study properties of the rank of the incidence matrix of these clutters and properties of their minors. A well known conjecture of Conforti and Cornu\'ejols \cite{ConfortiCornuejols,cornu-book} states: That all the clutters with the packing property have the max-flow min-cut property, i.e. are mengerian. Among the general classes of clutters known to verify the conjecture are: balanced clutters (Fulkerson, Hoffman and Oppenheim \cite{FulkersonHoffmanOppenheim}), binary clutters (Seymour \cite{Seymour}) and dyadic clutters (Cornu\'ejols, Guenin and Margot \cite{CornuejolsGueninMargot}). We find a new infinite family of 2-partitionable clutters, that verifies the conjecture. On the other hand we are interested in studying the normality of the Rees algebra associated to a clutter and possible relations with the Conforti and Cornu\'ejols conjecture. In fact this conjecture is equivalent to an algebraic statement about the normality of the Rees algebra \cite{rocky}.