Searcharxiv⌕ Search

arXiv subjects

Akihiro Shikama

Publications and source records attributed to Akihiro Shikama.

13 recordsLinked to original sources
arXiv2016

The numbers of edges of the order polytope and the chain poyltope of a finite partially ordered set

Let $P$ be an arbitrary finite partially ordered set. It will be proved that the number of edges of the order polytope ${\mathcal O}(P)$ is equal to that of the chain polytope ${\mathcal C}(P)$. Furthermore, it will be shown that the degree sequence of the finite simple graph which is the $1$-skeleton of ${\mathcal O}(P)$ is equal to that of ${\mathcal C}(P)$ if and only if ${\mathcal O}(P)$ and ${\mathcal C}(P)$ are unimodularly equivalent.

math.CO↗
arXiv2016

On the relations of isotonian algebras

It is shown that for large classes of posets $P$ and $Q$, the defining ideal $J_{P,Q}$ of an isotonian algebras is generated by squarefree binomials. Within these classes, those posets are classified for which $J_{P,Q}$ is quadratically generated.

math.AC↗
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

Isotonian Algebras

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.

math.AC↗
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↗
arXiv2015

Toric rings of nonsimple polyominoes

It is known that toric ring of a simple polyomino is ring homomorphic to a edge ring of a weakly chordal bipartite graph. In this paper we identify the toric ring of nonsimple polyominoes which are of the form "rectangle minus rectangle".

math.AC↗
arXiv2015

Simple Polyominoes are Prime

In this paper we show that polyomino ideal of a simple polyomino coincides with the toric ideal of a weakly chordal bipartite graph and hence it has a quadratic Gröbner basis with respect to a suitable monomial order.

math.AC↗
arXiv2014

Decomposable edge polytopes of finite graphs

Edge polytopes is a class of interesting polytope with rich algebraic and combinatorial properties, which was introduced by Ohsugi and Hibi. In this papar, we follow a previous study on cutting edge polytopes by Hibi, Li and Zhang. Instead of focusing on the algeraic properties of the subpolytopes as the previous study, in this paper, we take a closer look on the graphs whose edge polytopes are decomposable. In particular, we answer two important questions raised in the previous study about 1) the relationship between type I and type II decomposable graphs and 2) description of decomposable graphs.

math.CO↗
arXiv2014

Gröbner bases of balanced polyominoes

We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a quadratic Gröbner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced.

math.AC↗
arXiv2013

Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases

Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$ possesses no quadratic Gröbner basis. First, we give a nontrivial infinite series of finite graphs with the above property. Second, we implement a combinatorial characterization for $I_{G}$ to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs $G$ with the above property, up to 8 vertices.

math.AC↗