Graphs whose support-enumerators satisfy preorder conjectures

Characterize the bipartite graphs G for which the support-enumerator of the draconian-sequence set D(G) is unimodal, palindromic, and real-rooted, subject to the conjectural properties proposed for h-polynomials of preorders.

Background

The paper identifies the support-enumerator of D(G) with the h*-polynomial of an associated root polytope and notes that Chapoton and Athanasiadis conjectured unimodality, palindromicity, and real-rootedness for h-polynomials of preorders.

The authors observe that some of these properties, including palindromicity, fail for support-enumerators in general. The unresolved classification problem is therefore to determine precisely which bipartite graphs retain the conjectured properties.

References

In particular, they conjecture that the $h$-polynomial of a preorder is unimodal and palindromic, and only has real roots. It is easy to see that some of these conjectures, such as palindromicity, do not hold for support-enumerators in general. Nevertheless, this raises the following natural question: For which bipartite graphs $G$ do the support-enumerators of $D(G)$ satisfy the conjectures proposed in cite{chapoton2026}?

— Counting Lattice Points in Minkowski Sums of Cross Polytopes  (2608.16037 - Wang et al., 17 Aug 2026) in Section 4, Further Questions, third Problem