Powers of Edge Ideals with Linear Quotients

Abstract

We prove that second and higher powers of the edge ideals of anticycles admit linear quotient orderings, although the edge ideals themselves do not, thus resolving an open question of Hoefel and Whieldon in the affirmative and providing the first class of gap-free graphs whose edge ideals satisfy this property on their powers. We also construct an explicit and straightforward linear quotient ordering for any power of a quadratic monomial ideal which admits linear quotients. This expands on a well-known result of Herzog, Hibi, and Zheng. As a consequence, we give explicit formulas for the projective dimension and Betti numbers of the edge ideals of whisker graphs.

24 pages; edits to introduction, citations, and notation; removed unnecessary hypothesis in Theorem 3.11; added Remark 3.6