On 2-partitionable clutters and the MFMC property

We introduce 2-partitionable clutters as the simplest case of the class of kpartitionable 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 [1, 2] 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 [5]), binary clutters (Seymour [11]) and dyadic clutters (Cornuéjols, Guenin and Margot [3]). 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 [6].