Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.20430

Simplex faces and quadratic toric ideals of lattice polytopes

Aki Mori·Hidefumi Ohsugi
View original source ↗Read source document ↗Export metadata ↓

Abstract

We say that a convex polytope has the clique-face property if every clique in its 1-skeleton is the vertex set of a face. We establish this property as a geometric necessary condition for quadratic generation of toric ideals. More precisely, we prove that every lattice polytope with primitive edges and a quadratic toric ideal has the clique-face property; in particular, this holds for every $(0,1)$-polytope with a quadratic toric ideal. For $(0,1)$-polytopes satisfying condition (E), we characterize the clique-face property in terms of divisibility by monomials occurring in quadratic binomials, and show that, under the clique-face property, such toric ideals have no indispensable monomials of degree at least three. For edge polytopes and cut polytopes, we prove that the clique-face property is equivalent to quadratic generation. This yields new geometric characterizations of quadratic generation for these classes. We also prove that all simple polytopes, matroid independence polytopes, and matroid base polytopes have the clique-face property, and discuss the case of stable set polytopes in connection with conjectures on quadratic toric ideals.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Aki Mori, Hidefumi Ohsugi. 2026-07-22. Simplex faces and quadratic toric ideals of lattice polytopes. https://arxiv.org/abs/2606.20430

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗
arXiv2026

Random algebraic constructions for extremal and Ramsey problems

Building on Bukh's random algebraic method, we develop a framework for extremal and Ramsey problems involving apex hypergraphs. If $\mathcal{H}$ is a $(d-1)$-partite $(d-1)$-uniform hypergraph with $S$ edges and $\mathcal{H}(t)$ is obtained by adjoining $t$ vertices with common link $\mathcal{H}$, we prove that $\operatorname{ex}(n,\mathcal{H}(t))=Ω_{\mathcal{H}}(n^{d-1/S})$ for $t>9^{S+o_d(S)}$, which is best possible when $\mathcal{H}$ is Sidorenko. Our framework also yields sharper sided Zarankiewicz bounds, quantitative generalized Tur'an bounds, and diagonal multicolor Ramsey constructions. For each fixed $s\geq 2$ and $K\geq 3$, we further prove $\operatorname{r}_K(\mathcal K_{s,t};\mathcal K_n) =Θ_{s,t,K}((n/\log n)^s)$ for $t>9^{s+o(s)}$, extending a theorem of Alon and Rödl from factorial to exponential $t$. The main ingredients are interpolation on $m$-independent varieties, control of the dependencies imposed by symmetry, and linear spaces of forms whose nonzero members remain regular after a common algebraic slice. Limited edge independence then gives the spectral and local-density estimates needed for the Ramsey application.

math.CO↗
arXiv2026

A note on vertex-critical induced subgraphs of shift graphs

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a \emph{unique} induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

math.CO↗