Symmetric edge polytopes are not gamma-positive

Published 2 Jul 2026 in math.CO | (2607.02424v1)

Abstract: A conjecture posed by Ohsugi and Tsuchiya (2019) postulates that the Ehrhart $h^{*$-polynomials} of symmetric edge polytopes are $γ$-positive. We disprove this conjecture by exhibiting an infinite family of counterexamples. The smallest example provided by our construction is a $36$-dimensional symmetric edge polytope.

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Authors (1)

Luis Ferroni

Summary

The paper constructs an infinite family of series-parallel graph counterexamples that disprove gamma-positivity in symmetric edge polytopes.
It derives explicit h*-polynomial formulas demonstrating negative gamma coefficients, with the smallest counterexample in 36 dimensions.
The findings reveal that unimodality and reflexivity in Ehrhart theory do not imply gamma-positivity, urging refined conjectures.

Symmetric Edge Polytopes and the Failure of Gamma-Positivity

Introduction

The paper "Symmetric edge polytopes are not gamma-positive" (2607.02424) addresses a longstanding conjecture concerning the $\gamma$ -positivity of Ehrhart $h^*$ -polynomials for symmetric edge polytopes of graphs. These polytopes, defined as the convex hulls of vectors $\pm(e_i - e_j)$ associated to edges $ij$ in a graph $G$ with $n$ vertices, have been central objects in algebraic and geometric combinatorics, particularly via their connections to reflexive polytopes, unimodality conjectures, and Ehrhart theory.

A key conjecture, posed by Ohsugi and Tsuchiya, suggested that the Ehrhart $h^*$ -polynomials of symmetric edge polytopes are always $\gamma$ -positive. This conjecture was well-supported by empirical computations for small graphs and partial results establishing positivity for significant graph classes. The present paper disproves this conjecture by constructing an infinite family of counterexamples, specifically series-parallel graphs, and provides explicit calculations of $h^*$ and $\gamma$ -vectors demonstrating the presence of negative coefficients.

Background

Symmetric edge polytopes $h^*$ 0 possess several notable polyhedral properties relative to Ehrhart theory:

They are reflexive, implying that their $h^*$ 1-vectors are palindromic.
Their $h^*$ 2-polynomials are known to be unimodal due to regular unimodular triangulations (via Bruns and Römer [bruns-romer]).
For certain graph classes (e.g., cones over graphs and complete bipartite graphs), $h^*$ 3-positivity was established [higashitani-jochemko-michalek, ohsugi-tsuchiya-conj].

$h^*$ 4-positivity is a strengthening of unimodality for palindromic polynomials and has deep connections with topological combinatorics, originally formulated in the context of the Charney–Davis and Neggers–Stanley conjectures [gal, branden-gamma]. A palindromic polynomial $h^*$ 5 is $h^*$ 6-positive if it can be written as $h^*$ 7 with $h^*$ 8 for all $h^*$ 9.

Prior to this work, no counterexamples to the $\pm(e_i - e_j)$ 0-positivity conjecture had been identified among symmetric edge polytopes, and several results confirmed it for small graphs and special families [dali-juhnke-venturello, codenotti-riccardi-venturello, kalman-tothmeresz0, ohsugi-tsuchiya-2020-2].

Construction of Counterexamples

The paper constructs a family of series-parallel graphs $\pm(e_i - e_j)$ 1 with the following structure: select $\pm(e_i - e_j)$ 2 disjoint paths of lengths $\pm(e_i - e_j)$ 3 connecting two distinguished vertices $\pm(e_i - e_j)$ 4 and $\pm(e_i - e_j)$ 5. Counterexamples to $\pm(e_i - e_j)$ 6-positivity arise when taking $\pm(e_i - e_j)$ 7 sufficiently large and each path of sufficiently large length.

The explicit formula for the $\pm(e_i - e_j)$ 8-polynomial of the symmetric edge polytope $\pm(e_i - e_j)$ 9 is provided using the spanning-tree dissection technique as follows:

For fixed path length $ij$ 0 and number of paths $ij$ 1,

$ij$ 2

with $ij$ 3. The $ij$ 4-polynomial is given by

$ij$ 5

where

$ij$ 6

This explicit formula enables the computation of $ij$ 7- and $ij$ 8-coefficients for arbitrarily large graphs within this family.

Main Numerical Results

For the graph $ij$ 9, which forms the smallest explicit counterexample identified, the $G$ 0-dimensional $G$ 1-vector is: $G$ 2

The corresponding $G$ 3-vector is: $G$ 4

This provides a direct and explicit disproof of the Ohsugi–Tsuchiya conjecture: in this case, the $G$ 5-vector contains a negative coefficient ( $G$ 6). Furthermore, for even larger constructions (e.g., $G$ 7 paths of length $G$ 8), the distribution of signs in the $G$ 9-vector shows multiple negative coefficients, indicating that the violation of $n$ 0-positivity is neither isolated nor accidental but persistent in large dimensional cases.

Implications and Theoretical Consequences

The explicit construction of counterexamples to $n$ 1-positivity for symmetric edge polytopes definitively resolves the Ohsugi–Tsuchiya conjecture in the negative. This has several notable implications:

Limits of Positive Conjectures in Ehrhart Theory: Despite strong empirical support and partial cases, $n$ 2-positivity is not a universal feature of reflexive polytopes with palindromic and unimodal $n$ 3-vectors. For symmetric edge polytopes, regular unimodular triangulations and unimodality do not imply $n$ 4-positivity.
Series-Parallel Graphs as a Source of Complexity: The failure occurs in series-parallel graphs, which are among the most tractable classes in graph theory. This suggests that the combinatorial essence of the failure resides in the structural properties of these graphs and is unrelated to traditional graph-theoretic complexity.
Role of Combinatorial and Geometric Techniques: The paper leverages spanning-tree activities and carefully analyzes orientations and dual arborescences, underscoring the power of these combinatorial-geometric techniques for studying Ehrhart polynomials.
Future Research Directions: The failure of $n$ 5-positivity highlights the need to refine or replace existing conjectures by seeking structural graph properties that imply $n$ 6-positivity, or by better understanding how negative $n$ 7-coefficients arise. This could also influence analogous conjectures in the positively-graded cases of other classes of polytopes and matroid invariants [davis-higashitani-ohsugi, dali-juhnke-koch].

Conclusion

This work provides a rigorous negative resolution to a prominent conjecture in the combinatorial theory of lattice polytopes, establishing that symmetric edge polytopes need not have $n$ 8-positive Ehrhart $n$ 9-polynomials. The construction, using series-parallel graphs and explicit formulas, demonstrates the existence of infinite families of counterexamples. These findings refine our understanding of the interplay between combinatorics, geometry, and algebraic properties of polytopes and point toward a more nuanced landscape for future investigations of positivity phenomena in discrete geometry.