Buchsbaumness of the second powers of edge ideals
We graph-theoretically characterize the class of graphs $G$ such that $I(G)^2$ are Buchsbaum.
math.CO↗
arXiv subjects
Tran Nam Trung
Publications and source records attributed to Tran Nam Trung.
Regularity of powers of cover ideals of unimodular hypergraphs
Let $\H$ be a unimodular hypergraph over the vertex set $[n]$ and let $J(\H)$ be the cover ideal of $\H$ in the polynomial ring $R=K[x_1,\ldots,x_n]$. We show that $\reg J(\H)^s$ is a linear function in $s$ for all $s\geqslant r\left\lceil \frac{n}{2}\right\rceil+1$ where $r$ is the rank of $\H$. Moreover for every $i$, $a_i(R/J(\H)^s)$ is also a linear function in $s$ for $s \geqslant n^2$.
math.AC↗
Regularity of symbolic powers and Arboricity of matroids
Let $Δ$ be a simplicial complex of a matroid $M$. In this paper, we explicitly compute the regularity of all the symbolic powers of a Stanley-Reisner ideal $I_Δ$ in terms of combinatorial data of the matroid $M$. In order to do that, we provide a sharp bound between the arboricity of $M$ and the circumference of its dual $M^*$.
math.AC↗
Coverings, Matchings and the number of maximal independent sets of graphs
We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for König-Egerváry graphs in terms of their matching numbers.
math.CO↗
A Characterization of Gorenstein Planar graphs
We prove that a planar graph is Gorenstein if and only if its independence complex is Eulerian.
math.AC↗
Stability of Depths of Powers of Edge Ideals
Let $G$ be a graph and let $I := I (G)$ be its edge ideal. In this paper, we provide an upper bound of $n$ from which $\depth R/ I(G)^n$ is stationary, and compute this limit explicitly. This bound is always achieved if $G$ has no cycles of length $4$ and every its connected component is either a tree or a unicyclic graph.
math.AC↗
Depth and regularity of powers of sums of ideals
Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and $B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of $I$ and $J$. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.
math.AC↗
A Characterization of Triangle-free Gorenstein graphs and Cohen-Macaulayness of second powers of edge ideals
We graph-theoretically characterize triangle-free Gorenstein graphs $G$. As an application, we classify when $I(G)^2$ is Cohen-Macaulay.
math.AC↗
Regularity of powers of forests and cycles
Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I^s for all s > 0. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I^s) as s > 0, and the initial value of s starting from which reg(I^s) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.
math.AC↗
Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals
Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay). As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).
math.AC↗