Cycle algebras and polytopes of matroids

Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an algebraic and geometric investigation of these polytopes by studying their toric algebras, called cycle algebras, and their defining ideals. Several matroid operations are considered which determine faces of cycle polytopes that belong again to this class of polyhedral objects. As a key technique used in this paper, we study certain minors of given matroids which yield algebra retracts on the level of cycle algebras. In particular, that allows us to use a powerful algebraic machinery. As an application, we study highest possible degrees in minimal homogeneous systems of generators of defining ideals of cycle algebras as well as interesting cases of cut polytopes and Eulerian subgraph polytopes.

Diameter and connectivity of finite simple graphs

Let $G$ be a finite simple non-complete connected graph on ${1, ldots, n}$ and $kappa(G) geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $mathrm{diam}(G)$ the diameter of $G$. Being motivated by the computation of the depth of the binomial edge ideal of $G$, the possible sequences $(n, q, f, d)$ of integers for which there is a finite simple non-complete connected graph $G$ on ${1, ldots, n}$ with $q = kappa(G), f = f(G), d = mathrm{diam}(G)$ satisfying $f + d = n + 2 - q$ will be determined. Furthermore, finite simple non-complete connected graphs $G$ on ${1, ldots, n}$ satisfying $f(G) + mathrm{diam}(G) = n + 2 - kappa(G)$ will be classified.

Induced matchings in strongly biconvex graphs and some algebraic applications

In this paper, motivated by a question posed in cite{AH}, we introduce strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs. We give a linear time algorithm to find an induced matching for such graphs and we prove that this algorithm indeed gives a maximum induced matching. Applying this algorithm, we provide a strongly biconvex graph whose (monomial) edge ideal does not admit a unique extremal Betti number. Using this constructed graph, we provide an infinite family of the so-called closed graphs (also known as proper interval graphs) whose binomial edge ideals do not have a unique extremal Betti number. This, in particular, answers the aforementioned question in cite{AH}.

Binomial edge ideals of regularity $3$

Let $J_G$ be the binomial edge ideal of a graph $G$. We characterize all graphs whose binomial edge ideals, as well as their initial ideals, have regularity $3$. Consequently we characterize all graphs $G$ such that $J_G$ is extremal Gorenstein. Indeed, these characterizations are consequences of an explicit formula we obtain for the regularity of the binomial edge ideal of the join product of two graphs. Finally, by using our regularity formula, we discuss some open problems in the literature. In particular we disprove a conjecture in cite{CDI} on the regularity of weakly closed graphs.

The coordinate ring of a simple polyomino

In this paper it is shown that a polyomino is balanced if and only if it is simple. As a consequence one obtains that the coordinate ring of a simple polyomino is a normal Cohen-Macaulay domain.