Gapfree graphs and powers of edge ideals with linear quotients

2024/12/09 by Nursel Erey, Erey, Nursel, Sara Faridi +9
Computer Science · Mathematics · #05E40 #13D02 #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2412.06467

openalex publication_date 2024/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let I(G) be the edge ideal of a gapfree graph G. An open conjecture of Nevo and Peeva states that I(G)^q has linear resolution for q≫ 0. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if I(G)^q has linear quotients for some integer q≥ 1, then I(G)^s has linear quotients for all s≥ q. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if G does not contain a cricket, a diamond, or a C₄, then I(G)^q has linear resolution for q ≥ 2. We construct a family of gapfree graphs G containing cricket, diamond, C₄ together with C₅ as induced subgraphs of G for which I(G)^q has linear quotients for q ≥ 2.

Related

• Characterization of graphs whose a small power of their edge ideals has a linear free resolution
• Relation between regularity of powers of edge ideals and (im, reg)-invariant extension
• Linear resolution of connected graph ideals and their powers
• Powers of Edge Ideals with Linear Quotients
• Linear Quotients of the Square of the Edge Ideal of the Anticycle