It is known that a Markov basis of the binary graph model of a graph $G$ corresponds to a set of binomial generators of cut ideals $I_{widehat{G}}$ of the suspension $widehat{G}$ of $G$. In this paper, we give another application of cut ideals to sta
tistics. We show that a set of binomial generators of cut ideals is a Markov basis of some regular two-level fractional factorial design. As application, we give a Markov basis of degree 2 for designs defined by at most two relations.
Hara, Takemura and Yoshida discuss toric ideals arising from two way subtable sum problems and shows that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we
show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Groebner bases.
The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.
Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric
ideals $I_G$ of $G$ possesses a quadratic Grobner basis if the edge polytope ${cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed.
In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.
