Squarefree Powers of Edge Ideals of Forests

Abstract

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

Very well–covered graphs by Betti splittings

“…Then, using induction, we are able to prove that g I(G) is non-increasing (Theorem 2.1) supporting the expectation that g I is non-increasing for any squarefree monomial ideal I. We also compute the Castelnuovo-Mumford regularity reg(I(G) [k] ) in terms of the combinatorics of G (Theorem 2.13), solving affirmatively a conjecture due to Erey and Hibi [6,Conjecture 4.13].…”

“…Any tree posses at least two leaves. Let v ∈ V (G) be a leaf and w be the unique neighbor of v. Following [6], we say that v is a distant leaf if at most one of the neighbors of w is not a leaf. In this case, we say that {w, v} is a distant edge.…”

“…Squarefree powers have been introduced in [2], and are currently studied by many researchers [7,6,8,19,9,20].…”

On the Regularity of squarefree part of symbolic powers of edge ideals

“…In [7], Erey, Herzog, Hibi and Saeedi Madani initiated the study of regularity of squarefree part of powers of edge ideals. This study was continued in [8] and [25]. In this paper, we replace ordinary powers by symbolic powers and investigate the regularity of squarefree part of symbolic powers of edge ideals.…”

On the Castelnuovo-Mumford regularity of squarefree powers of edge ideals

“…In particular, inequality † is true for s = match(G). When G is a forest, Erey and Hibi [10] provided a sharp upper bound for reg(I(G) [s] ) in terms of the so-called s-admissable matching number of G. It follows from their result that inequality † is true for any forest.…”