Nonnegativity of the edge-deletion aggregate gamma-polynomial

Establish that for every 2-connected graph G, the polynomial obtained by summing the gamma-polynomials of h*_{P_G}(t)-h*_{P_{G\setminus e}}(t) over all edges e of G has nonnegative coefficients.

Background

For a 2-connected graph G and an edge e, deleting e preserves the dimension of the associated symmetric edge polytope, so the difference h*{P_G}(t)-h*{P_{G\setminus e}}(t) remains palindromic and has a gamma-polynomial. The proposed aggregate polynomial Z_G(t) sums these gamma-polynomials over all edges.

The conjecture is stronger than merely finding one edge whose deletion yields a gamma-nonnegative difference. The paper proves the quadratic-coefficient case and reports computational verification for all 2-connected graphs with at most eight vertices, but leaves coefficientwise nonnegativity unresolved.

References

Let $G=([n],E)$ be a $2$-connected graph. Then the polynomial $Z_G(t)$ has nonnegative coefficients.

— The number of edges of a symmetric edge polytope  (2512.16572 - Codenotti et al., 18 Dec 2025) in Conjecture 1.3, Section 1 (Introduction); restated as Conjecture in Section 3, subsection “Connection to z_2(G)”