# Conforti-Cornuéjols conjecture on the MFMC property

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 ${\mathcal Q}_6$ minor (one way to define ${\mathcal Q}_6$ is that the ground set is the edge set of $K_4$, and the hyperedges are the triangles). This implies the conjecture for binary clutters because ${\mathcal Q}_6$ 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.

## References

1. M. Conforti, G. Cornuéjols, Clutters that Pack and the Max Flow Min Cut Property: A Conjecture, Technical Report link
2. P.D. Seymour, The matroids with the max-flow min-cut property, DOI link
3. G. Cornuéjols, B. Guenin, F. Margot, The packing property, DOI link