Depth and Stanley depth of the edge ideals of the powers of paths and cycles

Abstract

Let k be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a path on n vertices. We show that both depth and Stanley depth have the same values and can be given in terms of k and n. If n≣0, k + 1, k + 2, . . . , 2k(mod(2k + 1)), then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the kth power of a cycle on n vertices and tight bounds otherwise, in terms of n and k. We also compute lower bounds for the Stanley depth of the edge ideals associated to the kth power of a path and a cycle and prove a conjecture of Herzog for these ideals.

Keywords

• monomial ideal
• edge ideal
• depth
• stanley decomposition
• stanley depth
• primary 13c15
• secondary 13f20
• 05c38
• 05e99