arXiv subjects

Aki Mori

Publications and source records attributed to Aki Mori.

10 recordsLinked to original sources

arXiv2026

Minkowski decomposability of symmetric edge polytopes

In this paper, we study the Minkowski decomposability of symmetric edge polytopes $P_G^\pm$ of a finite simple graph $G$ on vertex set $[n]$. More precisely, we give a complete characterization of graphs whose symmetric edge polytopes are Minkowski decomposable. We prove that $P_G^\pm$ is Minkowski decomposable if and only if $G$ is one of the three complete multipartite graphs: $K_n$, $K_{2,n-2}$, or $K_{1,1,n-2}$. In other words, if $G$ does not belong to these three families, then $P_G^\pm$ is Minkowski indecomposable.

math.CO↗

arXiv2026

Two-Dimensional Faces of Order and Chain Polytopes

We give an explicit combinatorial description of the two-dimensional faces of both the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a partially ordered set $P$. Using these descriptions, we show that for any $P$, $\mathcal{C}(P)$ has equally many square faces, and at least as many triangular faces, as $\mathcal{O}(P)$ does. Moreover, the inequality is shown to be strict except when $\mathcal{O}(P)$ and $\mathcal{C}(P)$ are unimodularly equivalent. This proves the case $i=2$ of a conjecture by Hibi and Li.

math.CO↗

arXiv2026

Facet numbers of non-centrally symmetric reflexive polytopes arising from posets

Twinned chain polytopes form a broad class of non-centrally symmetric reflexive polytopes and exhibit intriguing structures. In the present paper, we show that the number of facets of $d$-dimensional twinned chain polytopes is at most $6^{d/2}$. In case $d$ is even, the equality holds if and only if the polytope is isomorphic to a free sum of $d/2$ copies of del Pezzo polygons. This result contributes a partial answer to Nill's conjecture: the number of facets of a $d$-dimensional reflexive polytope is at most $6^{d/2}$.

math.CO↗

arXiv2026

Simplex faces and quadratic toric ideals of lattice polytopes

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.

math.CO↗

arXiv2025

Simplex faces of order and chain polytopes

It will be proved that a $k$-clique in the $1$-skeleton of either the order polytope or the chain polytope corresponds to the $(k-1)$-face, which is a simplex, in each polytope. These results generalize the known explicit descriptions of edges and triangular $2$-faces of each polytope.

math.CO↗

arXiv2025

Number of facets of symmetric edge polytopes arising from join graphs

Symmetric edge polytopes of graphs are important object in Ehrhart theory,and have an application to Kuramoto models. In the present paper, we study the upper and lower bounds for the number of facets of symmetric edge polytopes of connected graphs conjectured by Braun and Bruegge. In particular, we show that their conjecture is true for any graph that is the join of two graphs (equivalently, for any connected graph whose complement graph is not connected). It is known that any symmetric edge polytope is a centrally symmetric reflexive polytope. Hence our results give a partial answer to Nill's conjecture: the number of facets of a $d$-dimensional reflexive polytope is at most $6^{d/2}$.

math.CO↗

arXiv2025

Triangular faces of the order and chain polytope of a maximal ranked poset

Let $\mathscr{O}(P)$ and $\mathscr{C}(P)$ denote the order polytope and chain polytope, respectively, associated with a finite poset $P$. We prove the following result: if $P$ is a maximal ranked poset, then the number of triangular $2$-faces of $\mathscr{O}(P)$ is less than or equal to that of $\mathscr{C}(P)$, with equality holding if and only if $P$ does not contain an $X$-poset as a subposet.

math.CO↗

arXiv2022

The number of $4$-cycles and the cyclomatic number of a finite simple graph

Let $G$ be a finite connected simple graph with $n$ vertices and $m$ edges. We show that, when $G$ is not bipartite, the number of $4$-cycles contained in $G$ is at most $\binom{m-n+1}{2}$. We further provide a short combinatorial proof of the bound $\binom{m-n+2}{2}$ which holds for bipartite graphs.

math.CO↗

arXiv2016

The number of edges of the edge polytope of a finite simple graph

Let $d \geq 3$ be an integer. It is known that the number of edges of the edge polytope of the complete graph with $d$ vertices is $d(d-1)(d-2)/2$. In this paper, we study the maximum possible number $μ_d$ of edges of the edge polytope arising from finite simple graphs with $d$ vertices. We show that $μ_{d}=d(d-1)(d-2)/2$ if and only if $3 \leq d \leq 14$. In addition, we study the asymptotic behavior of $μ_d$. Tran--Ziegler gave a lower bound for $μ_d$ by constructing a random graph. We succeeded in improving this bound by constructing both a non-random graph and a random graph whose complement is bipartite.

math.CO↗

arXiv2015

Multibasic Ehrhart theory

In the present paper, we introduce a multibasic extension of the Ehrhart theory. We give a multibasic extension of Ehrhart polynomials and Ehrhart series. We also show that an analogue of Ehrhart reciprocity holds for multibasic Ehrhart polynomials.

math.CO↗