Related papers: Regularity 3 in edge ideals associated to bipartit…

In this paper, we mainly study the Castelnuovo-Mumford regularity of the generalized binomial edge ideals of graphs. We show that this number can be any integer number from $2$ to $n-1$ where $n$ is the number of vertices in the underlying…

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations.…

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free…

Binary classification problems can be naturally modeled as bipartite graphs, where we attempt to classify right nodes based on their left adjacencies. We consider the case of labeled bipartite graphs in which some labels and edges are not…

Let $G$ be a simple graph and $I_3(G)$ be its $3$-path ideal in the corresponding polynomial ring $R$. In this article, we prove that for an arbitrary graph $G$, $reg(R/I_3(G))$ is bounded below by $2\nu_3(G)$, where $\nu_3(G)$ denotes the…

An orientation of a graph $G$ is proper if any two adjacent vertices have different indegrees. The proper orientation number $\overrightarrow{\chi}(G)$ of a graph $G$ is the minimum of the maximum indegree, taken over all proper…

Consider the random process in which the edges of a graph $G$ are added one by one in a random order. A classical result states that if $G$ is the complete graph $K_{2n}$ or the complete bipartite graph $K_{n,n}$, then typically a perfect…

We study the topology of the lcm-lattice of edge ideals and derive upper bounds on the Castelnuovo-Mumford regularity of the ideals. In this context it is natural to restrict to the family of graphs with no induced 4-cycle in their…

Assume that $G$ is a graph with edge ideal $I(G)$. For every integer $s\geq 1$, we denote the squarefree part of the $s$-th symbolic power of $I(G)$ by $I(G)^{\{s\}}$. We determine an upper bound for the regularity of $I(G)^{\{s\}}$ when…

We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this…

We investigate the minimal free resolutions of closed neighborhood ideals of graphs within the framework of Barile-Macchia (BM) resolutions. We show that for any tree $T$, the closed neighborhood ideal $NI(T)$ is bridge-friendly, and hence…

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when…

The \emph{graph realization problem} is to find for given nonnegative integers $a_1,\dots,a_n$ a simple graph (no loops or multiple edges) such that each vertex $v_i$ has degree $a_i.$ Given pairs of nonnegative integers…

We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$…

It is known that complete graphs and complete multipartite graphs have modularity zero. We show that the least number of edges we may delete from the complete graph $K_n$ to obtain a graph with non-zero modularity is $\lfloor n/2\rfloor…

We define the Eulerian ideal of a $k$-uniform hypergraph and study its degree and Castelnuovo--Mumford regularity. The main tool is a Gr\"obner basis of the ideal obtained combinatorially from the hypergraph. We define the notion of parity…

A graph is regularizable if it is possible to assign weights to its edges so that all nodes have the same degree. Weights can be positive, nonnegative or arbitrary as soon as the regularization degree is not null. Positive and nonnegative…

Let $G$ be a simple graph with binomial edge ideal $J_G$. We prove how to calculate the multidegree of $J_G$ based on combinatorial properties of $G$. In particular, we study the set $S_{\min}(G)$ defined as the collection of subsets of…

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1,\ldots, y_n].$ We prove that the Castelnuovo-Mumford regularity of $J_G$ is bounded above by $c(G)+1$ when $G$…

In this paper we give upper bounds for the regularity of edge ideal of some classes of graphs in terms of invariants of graph. We introduce two numbers $a'(G)$ and $n(G)$ depending on graph $G$ and show that for a vertex decomposable graph…