Graded Betti numbers of path ideals of cycles and lines

Abstract

Abstract. We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of paths and cycles. As a consequence we can give new and short proofs for the known formulas of regularity and projective dimensions of path ideals of paths.

Projective dimension and regularity of edge ideals of some weighted oriented graphs

“…wm,t m m, tm ). We apply induction on m. The case m = 2 follows from Lemma 3.1 (2). Now we assume that m ≥ 3.…”

“…Note that the variables appearing in L 1 , L 2 and L i are different from each other. Therefore, it is enough to calculate reg (L 1 ) in order to compute reg ((J i : x w m,i+1 m, i+1 )) by Lemma 2.6 (2). We distinguish into the following two cases:…”

Projective dimension and regularity of path ideals of cycles

“…, F k . This settles Case (2). (c) If s j = 1 and r ≥ 2, since F 1 ∩ F h = ∅ for 1 < h ≤ k, and from (4.1) we obtain that for all i,…”

“…Proof. (1) can be shown by similar arguments as (2), so we only prove (2). According to Lemma 4.1, it is enough to show that this result is true for r = 1.…”

Regularity of path ideals of gap free graphs

“…The author of this paper in generalized Nevo's result in [3] by proving if G is gap free and cricket free (see definition) then I(G) n has linear minimal free resolution for every n ≥ 2. Compare to these, the case of the path ideal seems to be relatively less explored but significant works on the regularity of the path ideals have been done in some recent works (for example in [1], [2] and [4]). …”