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We introduce a family of graphs, which we call down-left graphs, and study their combinatorial and algebraic properties. We show that members of this family are well-covered, $C_5$-free, and vertex decomposable. By applying a result of…

Let $k$ be an arbitrary field, the purpose of this work is to provide families of positive integers $\mathcal{A} = \{d_1,\ldots,d_n\}$ such that either the toric ideal $I_{\mathcal A}$ of the affine monomial curve $\mathcal C =…

Let G be a finite simple graph. In this paper we will show that the binomial edge ideal of G, JG is toric if and only if each connected component of G is complete and in this case it is the sum of toric ideal associated to bipartite…

In this work, we classify the circuit binomials of any weighted oriented graph $D$ and we explicitly compute the circuit binomials of $D$ in terms of the minors of the incidence matrix of $D$. We show that the circuit binomials of any…

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there…

In this paper, we prove an analogue of Corr\'adi and Hajnal's classical theorem. There exists $n_0$ such that for every $n \in 3\mathbb{Z}$ when $n \ge n_0$ the following holds. If $G$ is an oriented graph on $n$ vertices and every vertex…

Given a set $\mathcal A = \{a_1,\ldots,a_n\} \subset \mathbb{N}^m$ of nonzero vectors defining a simplicial toric ideal $I_{\mathcal A} \subset k[x_1,...,x_n]$, where $k$ is an arbitrary field, we provide an algorithm for checking whether…

The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is…

Consider a graph $G$ and a $k$-uniform hypergraph $\mathcal{H}$ on common vertex set $[n]$. We say that $\mathcal{H}$ is $G$-intersecting if for every pair of edges in $X,Y \in \mathcal{H}$ there are vertices $x \in X$ and $y \in Y$ such…

Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph whose edge ideal is $I(D)$. In this paper, we characterize the unmixed property of $I(D)$ for each one of the following cases: $G$ is an $SCQ$ graph; $G$ is a chordal graph; $G$ is a…

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is…

A graph-theoretic method, simpler than existing ones, is used to characterize the minimal set of monomial generators for the integral closure of any algebra of polynomials generated by quadratic monomials. The toric ideal of relations…

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$)…

Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the…

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma, G)$ where $\Gamma$ is finite, 4-valent, connected, and $G$-oriented ($G$-half-arc-transitive). A subfamily of $\mathcal{OG}(4)$ has recently been identified as…

Let $G$ be a simple graph on the vertex set $\{v_{1},\ldots,v_{n}\}$. An algebraic object attached to $G$ is the toric ideal $I_G$. We say that $I_G$ is subgraph splittable if there exist subgraphs $G_1$ and $G_2$ of $G$ such that…

The total graph of $G$, $\mathcal T(G)$ is the graph whose set of vertices is the union of the sets of vertices and edges of $G$, where two vertices are adjacent if and only if they stand for either incident or adjacent elements in $G$. Let…

A graph is circle if there is a family of chords in a circle such that two vertices are adjacent if the corresponding chords cross each other. There are diverse characterizations of circle graphs, many of them using the notions of local…

If $G$ is a graph and $\vec H$ is an oriented graph, we write $G\to \vec H$ to say that every orientation of the edges of $G$ contains $\vec H$ as a subdigraph. We consider the case in which $G=G(n,p)$, the binomial random graph. We…

The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple undirected graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if their intersection is nontrivial.…