Characterization of graphs whose a small power of their edge ideals has a linear free resolution

2022/08/29 by Nguyên Công Minh, Minh, Nguyen Cong, Thanh Vu +1
Mathematics · Neuroscience · #05E40 #13D02 #13F55 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Mind wandering and attention #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2208.13745

openalex publication_date 2022/08/29 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28

Abstract

Let I(G) be the edge ideal of a simple graph G. We prove that I(G)² has a linear free resolution if and only if G is gap-free and regI(G) ≤ 3. Similarly, we show that I(G)³ has a linear free resolution if and only if G is gap-free and regI(G) ≤ 4. We deduce these characterizations from a general formula for the regularity of powers of edge ideals of gap-free graphs \rm reg(I(G)^s) = max(\rm reg I(G) + s-1,2s), for s =2,3.

Related

• Linear resolution of connected graph ideals and their powers
• Gapfree graphs and powers of edge ideals with linear quotients
• Relation between regularity of powers of edge ideals and (im, reg)-invariant extension
• Regularity of Path Ideals of Gap Free Graphs
• An upper bound for the regularity of powers of edge ideals