Ohsugi–Tsuchiya Conjecture

• Ohsugi–Tsuchiya Conjecture is a set of conjectures stating that the γ-vector in symmetric edge polytopes is nonnegative and that enriched poset polytopes share the same Ehrhart polynomial.
• It employs combinatorial, algebraic, and geometric techniques, including unimodular triangulations and Gröbner basis methods, to analyze polytopes derived from graphs, posets, and regular matroids.
• The conjecture reveals practical insights by showing that while graphic cases support γ-nonnegativity, counterexamples from non-graphic regular matroids prompt further study into circuit structures and triangulation strategies.

The Ohsugi–Tsuchiya Conjecture refers to two central conjectures in the combinatorial theory of lattice polytopes: the γ-nonnegativity conjecture for the symmetric edge polytope (SEP) and the Ehrhart-equivalence conjecture between enriched poset polytopes. Both conjectures illuminate deep combinatorial, algebraic, and geometric properties of polytopes associated to combinatorial objects such as graphs, posets, and regular matroids.

1. Symmetric Edge Polytopes and the γ-Vector

Let $G$ be a finite simple graph with vertex set of size $n$, and define the symmetric edge polytope (SEP) by

$$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$

where $e_i$ denotes the $i$th standard basis vector. The Ehrhart function $L_{\Sigma(G)}(m) = |m\,\Sigma(G)\cap \Z^n|$ has generating function

$$E(\Sigma(G); t) = \sum_{m\ge0} L_{\Sigma(G)}(m)t^m = \frac{h_0^* + h_1^* t + \cdots + h_d^* t^d}{(1-t)^n},$$

where $d = \dim \Sigma(G) = n-1$. The $h^*$-polynomial $h^*(\Sigma(G); t) = \sum_{i=0}^d h_i^* t^i$ is symmetric ($n$0) due to reflexivity and can be written as

$n$1

where $n$2 is called the γ-vector.

The Ohsugi–Tsuchiya Conjecture posits that for every finite graph $n$3, this γ-vector is entrywise nonnegative:

$n$4

2. Known Results and Extensions

Prior to recent work, γ-nonnegativity was established for various graph classes:

• Complete graphs $n$5, via flag, unimodular, anti-blocking triangulations and Gal's theorem.
• Complete bipartite graphs $n$6, encompassing trees and star graphs.
• Even cycles $n$7.
• Broader graph families admitting “locally anti-blocking” decompositions, as demonstrated by Ohsugi–Tsuchiya (2021), covering chordal and certain bipartite graphs.

Additionally, extensive computation on small graphs found no counterexamples to γ-nonnegativity. However, there was no explicit combinatorial formula for general γ-vectors, nor a universal proof mechanism outside known triangulation results (Davis et al., 2024).

3. Extension to Regular Matroids and Generalized SEPs

The notion of SEPs extends naturally to regular matroids—matroids representable by totally unimodular matrices. Given a regular matroid $n$8 with ground set $n$9 and rank $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$0 represented by a matrix $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$1 (with columns $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$2), the generalized SEP is

$$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$3

well-defined up to unimodular equivalence and of dimension $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$4. In the case where $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$5 is graphic, $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$6 coincides with the standard SEP. The $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$7-polynomial retains the reflective symmetry, allowing a γ-vector expansion in this broader setting (Davis et al., 2024).

4. Combinatorial and Gröbner Basis Techniques

Unimodular triangulations of $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$8 in the graphical setting are constructed by “cones over” oriented spanning trees. Kálmán–Tóthmérész (2023) provided an interpretation where $$\Sigma(G) = \mathrm{conv}\left\{ \pm(e_i - e_j) \mid ij \in E(G) \right\} \subset \R^n,$$9 counts spanning trees with $e_i$0 active edges, extendable to “basis” objects in regular matroids.

In the regular matroid framework, the associated toric ideal $e_i$1 in the coordinate ring $e_i$2 has a reduced Gröbner basis governed by the circuits of $e_i$3, with square-free leading monomials corresponding to faces of a regular unimodular triangulation. Counting “active” elements in these bases reconstructs the $e_i$4-vector and, thus, the γ-vector (Davis et al., 2024).

5. Counterexamples outside the Graphical Case and Near-Positivity

Explicit counterexamples occur for generalized SEPs:

• For the cographic matroid $e_i$5, which is regular but non-graphic when $e_i$6, direct computation for $e_i$7 yields a γ-vector $e_i$8, thereby violating γ-nonnegativity.
• These negative entries persist for higher dimensions via direct sums with loop-coloop matroids.

However, nearly γ-nonnegative behavior appears: by deleting two elements from $e_i$9, one obtains a graphic matroid for which the ordinary SEP recovers nonnegativity. The explicit computation for the special graph $i$0 exhibits γ-polynomials with manifestly nonnegative coefficients, via a closed-form expansion provided in the data.

6. The Enriched Poset Polytope Conjecture and Bijective Proof

An independent line of research by Ohsugi and Tsuchiya concerns Ehrhart-equivalence between the enriched order polytope $i$1 and the enriched chain polytope $i$2, defined for a finite poset $i$3 as the convex hulls of integer functions on $i$4 obeying specific filter and antichain constraints, respectively.

The Ohsugi–Tsuchiya Conjecture in this context states that their Ehrhart polynomials coincide. Okada and Tsuchiya constructed a piecewise-linear bijection—the enriched transfer map $i$5—inductively on the poset order, with explicit inverse, restricting to a bijection between $i$6 and $i$7 that preserves lattice points for all positive dilations:

$i$8

This bijection provides a combinatorial (triangulation-based) proof of the Ehrhart-equivalence conjecture. Both polytopes admit canonical unimodular triangulations, and the bijection $i$9 transports the triangulation of $L_{\Sigma(G)}(m) = |m\,\Sigma(G)\cap \Z^n|$0 onto that of $L_{\Sigma(G)}(m) = |m\,\Sigma(G)\cap \Z^n|$1 face-by-face (Okada et al., 2020).

7. Implications, Conditions, and Open Problems

For SEPs, the existence of negative γ-entries in the regular matroid setting underscores the necessity of the graphic hypothesis for the Ohsugi–Tsuchiya γ-nonnegativity conjecture. The fact that simple deletions can restore nonnegativity suggests that the obstructions reside in the extra flexibility of non-graphic matroids, particularly their expanded circuit structure.

In the field of enriched poset polytopes, the existence of a canonical bijection and compatible triangulations situates these objects within the broader context of mirror symmetry, mutation-equivalence, and Gröbner degenerations of Hibi rings.

Open questions include:

• Whether there exists a purely combinatorial, triangulation-free explanation of γ-nonnegativity in SEPs, potentially relating γ-coefficients to explicit counts of flagged combinatorial structures;
• Whether the $L_{\Sigma(G)}(m) = |m\,\Sigma(G)\cap \Z^n|$2-polynomial of SEP exhibits real-rootedness or stability properties, which would imply γ-nonnegativity;
• Whether there exist non-graphic regular matroids whose SEPs are γ-nonnegative, or whether such behavior is strictly forbidden by circuit structure.

The Ohsugi–Tsuchiya Conjecture continues to frame research into the intersection of combinatorics, discrete geometry, and commutative algebra, with known results robust in the graph-SEP case, but delicately sensitive to generalizations beyond graphic matroids (Davis et al., 2024, Okada et al., 2020).