On the Resolution of Path Ideals of Cycles

Abstract

We give a formula to compute all the top degree graded Betti numbers of the path ideal of a cycle. Also we will find a criterion to determine when Betti numbers of this ideal are non zero and give a formula to compute its projective dimension and regularity.

Regularity of Edge Ideals and Their Powers

“…Also bipartite graphs contain no closed circuit of length 5. Thus, in Question 7.9, an affirmative answer to part (4) would imply that for part (3).…”

Projective dimension and regularity of path ideals of cycles

“…, F k is the path complex of the cycle C n with standard labeling and I m,l (C n ) = I(∆ m,l (C n )) is the facet ideal of ∆ m,l (C n ). We can write k as k = p(t + 1) + d, where p ≥ 0 and 0 ≤ d ≤ t. Then (1) The projective dimension of R/I m,l (C n ) is given by pd (R/I m,l (C n )) = 2p, if d = 0 2p + 1, if d = 0…”

A constructive approach to minimal free resolutions of path ideals of trees

“…We will use a corollary of Theorem 2.8 of Alilooee and Faridi in [1] to help determine the multi-graded Betti numbers for path ideals of rooted trees. This corollary, found below, appears in [2] as Corollary 2.11.…”

“…The basic tools and decompositions that will be used in this paper were introduced by Faridi and Alilooee in [1]. The aim of this paper is to give a constructive description of the multi-graded Betti numbers for path ideals of rooted trees.…”