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Ohsugi–Tsuchiya gamma-nonnegativity conjecture for symmetric edge polytopes

Let GG be a graph, and let PG\mathcal{P}_G be its symmetric edge polytope. Write γ(PG)=(γ0,γ1,…)\gamma(\mathcal{P}_G)=(\gamma_0,\gamma_1,\ldots) for the γ\gamma-vector associated with the palindromic h∗h^*-vector of PG\mathcal{P}_G.

Ohsugi–Tsuchiya conjecture. For every i≥0i\geq 0,

γi(PG)≥0.\gamma_i(\mathcal{P}_G)\geq 0.

This conjecture extends gamma-nonnegativity from flag simplicial spheres to symmetric edge polytopes, whose relevant triangulations need not be flag. The supplied status evidence says that this instance was proved in the original work of Hibi, Matsuda, and Murai and generalized by Ohsugi and Tsuchiya to the indicated bipartite-graph setting; it is therefore solved.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Ohsugi–Tsuchiya gamma-nonnegativity conjecture for symmetric edge polytopes

    Let GG be a graph, let PGP_G be its symmetric edge polytope, and write its h∗h^*-polynomial in the palindromic basis as hPG∗(t)=∑j=0⌊d/2⌋γjtjh^*_{P_G}(t)=\sum_{j=0}^{\lfloor d/2\rfloor}\gamma_jt^j, where d=dim⁡PGd=\dim P_G. The associated γ\gamma-polynomial is γPG(t)=∑j=0⌊d/2⌋γjtj\gamma_{P_G}(t)=\sum_{j=0}^{\lfloor d/2\rfloor}\gamma_jt^j. Ohsugi–Tsuchiya's conjecture. The polynomial γPG(t)\gamma_{P_G}(t) has nonnegative coefficients for every graph GG. This conjecture concerns coefficient inequalities in the Ehrhart theory of symmetric edge polytopes; the supplied text gives no resolution status.

    source: Giulia Codenotti, Roberto Riccardi and Lorenzo Venturello, “The number of edges of a symmetric edge polytope”, arXiv:2512.16572 (2026).

Sources & referencesView supporting material

Primary source

Alessio D'Alì, Martina Juhnke-Kubitzke, Daniel Köhne and Lorenzo Venturello, “On the gamma-vector of symmetric edge polytopes”, arXiv:2201.09835 (2022).

ehrhart-theorygamma-positivitygamma-vectorsgraph-polytopessymmetric-edge-polytopes

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