# Conforti-Cornuéjols conjecture on the MFMC property

From Egres Open

Is it true that a clutter has the MFMC property if and only if it has the packing property?

## Remarks

This was conjectured by Conforti and Cornuéjols ^{[1]}. Some classes where the conjecture is known to hold:

- A clutter is
**binary**if every member of its blocker intersects each member of the clutter in an odd number of elements. Seymour^{[2]}proved that an ideal binary clutter has the MFMC propery if and only if it does not have a [math]{\mathcal Q}_6[/math] minor (one way to define [math]{\mathcal Q}_6[/math] is that the ground set is the edge set of [math]K_4[/math], and the hyperedges are the triangles). This implies the conjecture for binary clutters because [math]{\mathcal Q}_6[/math] does not have the packing property. - A clutter is
**diadic**if every member of its blocker intersects each member of the clutter in at most 2 elements. It is shown in^{[3]}that a diadic clutter is ideal if and only if it has the MFMC property. This implies the conjecture because every clutter with the packing property is ideal.