Regularity of squarefree monomial ideals

We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.Comment: 23 pages; survey paper; minor changes in V.

Fully dynamic maintenance of Euclidean minimum spanning trees and maxima of decomposable functions

We maintain the minimum spanning tree of a point set in the plane, subject to point insertions and deletions, in time O(n^1/2 log^2 n) per update operation. We reduce the problem to maintaining bichromatic closest pairs, which we solve in time O(n^E) per update. Our algorithm uses a novel construction, the ordered nearest neighbors of a sequence of points. Any point set or bichromatic point set can be ordered so that this graph is a simple path. Our results generalize to higher dimensions, and to fully dynamic algorithms for maintaining maxima of decomposable functions, including the diameter of a point set and the bichromatic farthest pair

Vertex decomposable graphs, codismantlability, Cohen-Macaulayness and Castelnuovo-Mumford regularity

We call a (simple) graph G codismantlable if either it has no edges or else it has a codominated vertex x, meaning that the closed neighborhood of x contains that of one of its neighbor, such that G-x codismantlable. We prove that if G is well-covered and it lacks induced cycles of length four, five and seven, than the vertex decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. The rest deals with the computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note that our approach complements and unifies many of the earlier results on bipartite, chordal and very well-covered graphs

Decomposing 1-Sperner hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper

Regularity of Edge Ideals and Their Powers

We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{ reg } I(G)$ and the asymptotic linear function $\text{ reg } I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$Comment: 31 pages, 15 figure

On the longest path in a recursively partitionable graph

A connected graph $G$ with order $n \geq 1$ is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to $K_1$, or for every sequence $(n_1, \ldots , n_p)$ of positive integers summing up to $n$ there exists a partition $(V_1, \ldots , V_p)$ of $V(G)$ such that each $V_i$ induces a connected R-AP subgraph of $G$ on $n_i$ vertices. Since previous investigations, it is believed that a R-AP graph should be 'almost traceable' somehow. We first show that the longest path of a R-AP graph on $n$ vertices is not constantly lower than $n$ for every $n$. This is done by exhibiting a graph family $\mathcal{C}$ such that, for every positive constant $c \geq 1$, there is a R-AP graph in $\mathcal{C}$ that has arbitrary order $n$ and whose longest path has order $n-c$. We then investigate the largest positive constant $c' \lt 1$ such that every R-AP graph on $n$ vertices has its longest path passing through $n \cdot c'$ vertices. In particular, we show that $c' \leq \frac{2}{3}$. This result holds for R-AP graphs with arbitrary connectivity

Sampling decomposable graphs using a Markov chain on junction trees

Full Bayesian computational inference for model determination in undirected graphical models is currently restricted to decomposable graphs, except for problems of very small scale. In this paper we develop new, more efficient methodology for such inference, by making two contributions to the computational geometry of decomposable graphs. The first of these provides sufficient conditions under which it is possible to completely connect two disconnected complete subsets of vertices, or perform the reverse procedure, yet maintain decomposability of the graph. The second is a new Markov chain Monte Carlo sampler for arbitrary positive distributions on decomposable graphs, taking a junction tree representing the graph as its state variable. The resulting methodology is illustrated with numerical experiments on three specific models.Comment: 22 pages, 7 figures, 1 table. V2 as V1 except that Fig 1 was corrected. V3 has significant edits, dropping some figures and including additional examples and a discussion of the non-decomposable case. V4 is further edited following review, and includes additional reference

Componentwise Linearity of Powers of Cover Ideals

Let $G$ be a finite simple graph and $J(G)$ denote its vertex cover ideal in a polynomial ring over a field. Assume that $J(G)^{(k)}$ is its $k$-th symbolic power. In this paper, we give a criteria for cover ideals of vertex decomposable graphs to have the property that all their symbolic powers are not componentwise linear. Also, we give a necessary and sufficient condition on $G$ so that $J(G)^{(k)}$ is a componentwise linear ideal for some (equivalently, for all) $k \geq 2$ when $G$ is a graph such that $G \setminus N_G[A]$ has a simplicial vertex for any independent set $A$ of $G$. Using this result, we prove that $J(G)^{(k)}$ is a componentwise linear ideal for several classes of graphs for all $k \geq 2$. In particular, if $G$ is a bipartite graph, then $J(G)$ is a componentwise linear ideal if and only if $J(G)^k$ is a componentwise linear ideal for some (equivalently, for all) $k \geq 2$.Comment: arXiv admin note: text overlap with arXiv:1908.1057