On 2-partitionable clutters and the MFMC property

Alejandro Flores-Méndez; Isidoro Gitler; Enrique Reyes

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arXiv · 0806.1772

On 2-partitionable clutters and the MFMC property

Alejandro Flores-Méndez·Isidoro Gitler·Enrique Reyes

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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 Cornuéjols \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 (Cornuéjols, 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 Cornuéjols conjecture. In fact this conjecture is equivalent to an algebraic statement about the normality of the Rees algebra \cite{rocky}.

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Alejandro Flores-Méndez, Isidoro Gitler, Enrique Reyes. 2008-06-11. On 2-partitionable clutters and the MFMC property. https://arxiv.org/abs/0806.1772

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