Saturation and associated primes of powers of edge ideals

Combinatorial Characterizations of the Saturation and the Associated Primes of the Fourth Power of Edge Ideals

“…So we can assume that Ω is connected. Due to [10,Lemma 2.7], Ω contains either a triangle, or a pentagon, or a C 7 . If Ω contains a C 7 , then Ω is of type 8.…”

“…We give all types of subgraphs such that every associated prime of I 4 corresponds to a cover of Γ which is minimal among the covers containing the closed neighborhood of such a subgraph of Γ. Together with the results of [13], [10], our results may be used to find combinatorial patterns of the associated primes of every power I t and to study the function depth R/I t .…”

“…For convenience set I = I(Γ). It has been shown in [10] that x a G ∈ (I t ) G \(I t ) G if and only if x a U ∈ J t \J t for some subset U ⊆ V , where J is the edge ideal of the reduced subgraph of Γ on V \ U. Therefore, to compute the local cohomology modules of R/I t we need to know the saturation of the t-th powers of edge ideals.…”

“…By [10], one can associate with such a monomial a vertex weighted graph Γ a with certain good matching properties, and one can describe all such weighted graphs for t = 3. From this it follows that there are five types of subgraphs such that every associated prime of I 3 corresponds to a cover of Γ which is minimal among the covers containing the closed neighborhood of such a subgraph of Γ.…”

“…In this paper, we use the method of [10] to solve the above problem for t = 4 and we give a complete classification of the associated primes of I 4 . We give all types of subgraphs such that every associated prime of I 4 corresponds to a cover of Γ which is minimal among the covers containing the closed neighborhood of such a subgraph of Γ.…”