Symbolic powers of cover ideal of very well-covered and bipartite graphs

Abstract

Let G be a graph with n vertices and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over a field K. Assume that J(G) is the cover ideal of G and J(G) (k) is its k-th symbolic power. We prove that if G is a very well-covered graph such that J(G) has linear resolution, then J(G) (k) has linear resolution, for every integer k ≥ 1. We also prove that for a every very well-covered graph G, the depth of symbolic powers of J(G) forms a non-increasing sequence. Finally, we determine a linear upper bound for the regularity of powers of cover ideal of bipartite graph.

On the minimal free resolution of symbolic powers of cover ideals of graphs

“…Thus, the result of Herzog and Hibi essentially says that if G is a bipartite graph such that J(G) has a linear resolution, then J(G) (k) has a linear resolution, for every integer k ≥ 1. In [16,Theorem 3.6], we generalized this result to very well-covered graphs. In the same paper, we posed the following question.…”

“…In the same paper, we posed the following question. Question 1.1 ( [16], Page 105 and [19], Question 3.10). Let G be a very well-covered graph and suppose that J(G) (k) has a linear resolution, for some integer k ≥ 2.…”

“…In [16,Corollary 3.7], we proved that Question 1.1 has a positive answer for the special class of bipartite graphs. As the first main result of this paper, we give a positive answer to this question, for any very well-covered graph (see Proposition 3.1).…”

Induced Matchings and the v-Number of Graded Ideals

“…The projective dimension of the edge ideal of a graph, the Wiener index, the independence polynomial, the h-vector, and the symbolic powers of cover ideals of graphs have been studied for very well-covered graphs [45][46][47][48][49][50][51].…”

Depth and regularity modulo a principal ideal

“…The main part of this work was done during research stays of the authors at the American Institute of Mathematics in the SQuaRE program "Ordinary powers and symbolic powers" during the period 2012-2014. The authors are grateful to S. A. Seyed Fakhari for pointing out a mistake of Theorem 5.1 in the first version of this paper and to an anonymous referee for mentioning that Lemma 4.1(i) and (ii) were already proved in [30] and [31], respectively.…”