On the Arithmetical Rank of the Edge Ideals of Forests

Abstract

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.

Lyubeznik resolutions and the arithmetical rank of monomial ideals

“…The author together with Terai and Yoshida ( [9,10]; see also [11]) has proved that ara I = pd S S/I for squarefree monomial ideals I with µ(I) − height I ≤ 2. Barile [3,4,5,6,7], Barile and Terai [8], and Schmitt and Vogel [15] also proved the same equality for some classes of squarefree monomial ideals. Since the projective dimension of S/I is equal to the length of the minimal graded free resolution of S/I, we have the following corollary: In Section 2, we prove Theorem 1 and several corollaries.…”

Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs

“…Furthermore, ( [3], p. 4701) implies that araI(L 3t j +2 ) = 2t j + 1 and araI(L 3s i ) = 2s i , and hence…”

“…n i ≡ 0 (mod 3) for any 1 ≤ i ≤ k; 2. there exists exactly one n j such that n j ≡ 1 (mod 3), and for any 1 ≤ i = j ≤ k, we have n i ≡ 2 (mod 3).…”

“…For the edge ideal of a forest, it is shown that bight(I(G)) = araI(G) = pd(R/I(G)) by Barile [3] and Kimura and Terai [4]. In [5], Barile et al proved that araI(G) = pd(R/I(G)) when G is a cyclic or bicyclic graph.…”

Arithmetical rank of the edge ideals of some $n$-cyclic graphs with a common edge

“…Thus, I is generated by square-free quadratic monomials and is therefore a radical ideal. The problem of the arithmetical rank of edge ideals or monomial ideals has been intensively studied by many authors over the past 3 decades (see [1,2,3,5,8,10]). …”