The regularity of powers of edge ideals

Abstract

Abstract. In this paper we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order we find new upper bounds on the regularity of powers of edge ideals of graphs whose complement does not have any induced four cycle.

Regularity of path ideals of gap free graphs

“…We prove various results of this type in the cases where there are results about the edge ideals, for example the gap free graphs, the claw free graphs and the cricket free graphs. Our main result is the following (compare this to the work done in [3] or [13]): Theorem 1.1. If G is gap free and claw free then for every t ≥ 3 the squarefree monomial ideal generated by t-paths I t (G) is either the zero ideal or has linear minimal free resolution.…”

“…Using this lemma repeatedly we get the following result, which is also a varsion of the Lemma 5.1 of [3]: Lemma 2.10. Let J ⊆ I be two monomial ideals in the polynomial ring S and I is generated in degree d by m 1 , ..., m k .…”

“…We refer reader to [3] and [6] for reference. Moreover, if m is a variable x appearing in I, then reg(I) is equal to one of these terms.…”

“…He proved that if G is both gap free and claw free then I(G) has regularity less than or equal to 3 and I(G) 2 has linear resolution. 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]).…”

Regularity of powers of edge Ideal of whiskered cycles

“…Computing and finding bounds for the regularity of edge ideals and their powers have been studied by a number of researchers (see for example [1], [2], [3], [5], [7], [11] and [12]). It is well-known that reg(I s ) is asymptotically a linear function for s ≫ 0.…”