Conjectured formula for h*-polynomial differences under edge deletion

Derive the stated decomposition of h*_{P_G}(t)-h*_{P_{G\setminus ij}}(t) for every 2-connected graph G and edge ij, and prove that each summand in the decomposition is palindromic with nonnegative coefficients and has the same center as the full difference.

Background

The paper derives a formula for the h*-polynomial difference after deleting an edge using a triangulation of the facets visible from the deleted edge. The resulting decomposition has summands with nonnegative coefficients, but those summands are not generally palindromic in the formula already proved.

The authors propose a second, conjectural decomposition indexed by lattice distances from the deleted edge to visible facets. If valid, it would express the difference as a sum of palindromic pieces and would provide a way to study how gamma-positivity changes under edge deletion. The formula is verified in several examples and in the special case where all visible facets have lattice distance one.

References

The following conjecture predicts that we can write $h*_{P_G}(t) - h*{P{G ij}(t)$ as the sum of palindromic polynomials in terms of these subcomplexes.

— The number of edges of a symmetric edge polytope  (2512.16572 - Codenotti et al., 18 Dec 2025) in Conjecture 4.1, Section 4 (Further directions)