Linear Quotients of the Square of the Edge Ideal of the Anticycle

Abstract: Let $G$ be a graph with chordal complement and $I(G)$ its edge ideal. From work of Herzog, Hibi, and Zheng, it is known that $I(G)$ has linear quotients and all of its powers have linear resolutions. For edge ideals $I(G)$ arising from graphs which do not have chordal complements, exact conditions on their powers possessing linear resolutions or linear quotients are harder to find. We provide here an explicit linear quotients ordering for all powers of the edge ideal of the antipath and a linear quotients ordering on the second power of $I(A_n)^2$ of the edge ideal of the anticycle $A_n$. This linear quotients ordering on $I(A_n)$ recovers a prior result of Nevo that $I(A_n)^2$ has a linear resolution.