# Characterizing integral g-polymatroids

Pritchard conjectures the following^{[1]} about *g*-polymatroids:

For any polyhedron [math]P[/math] which is not an integral *g*-polymatroid, there is an integral *g*-polymatroid [math]Q[/math] such that [math]P \cap Q[/math] is non-integral.

The conjecture was proved by Júlia Pap ^{[2]} using a characterization of g-polymatroids by Tomizawa.

## Remarks

In the paper introducing *g*-polymatroids^{[3]}, Frank showed that the intersection of any two integral polymatroids is integral. (As usual, a polyhedron is *integral* if every face contains an integer vector.) Thus, the conjecture is a converse to Frank's theorem.

Since [math]R^n[/math] is a *g*-polymatroid, the theorem is trivial if *P* is not integral, since we can take [math]Q=R^n[/math]. Therefore we may assume *P* is integral, but not a *g*-polymatroid.

## References

- ↑ D. Pritchard.
*The Hypergraphic Tutte/Nash-Williams Theorem via Integer Decomposition, Total Dual Laminarity, and Power Matroids*, EGRES Quick Proof no. 2010-06. - ↑ J. Pap,
*A note on generalized polymatroids*, EGRES Quick Proof no. 2011-03. - ↑ A. Frank.
*Generalized polymatroids.*In A. Hajnal, L. Lovász, and V. T. Sós, editors, Finite and Infinite Sets, Vol. I (Eger 1981), volume 37 of Colloq. Math. Soc. János Bolyai, pages 285–294. North-Holland, 1984. Author link.