Date/Time
Date(s) - 28/10/2025
11:30 am - 12:30 pm

Location: HH 312

Speaker: Selvi Kara (Bryn Mawr)

Title: Gapfree graphs and powers of their edge ideals

Abstract: Powers of edge ideals often reflect the combinatorics of the underlying graph. Let $G$ be a graph and let $I(G)$ be its edge ideal. For gapfree graphs, a conjecture of Nevo–Peeva states that high enough powers of the edge ideal have linear resolutions. In this talk, I will describe our approach to this conjecture via a new conjecture involving linear quotients. Our conjecture is monotone: once one power $I(G)^q$ has linear quotients, then every higher power $I(G)^s$ (for $s\geq q$) should as well. I’ll present partial progress, including hypotheses under which the problem reduces to checking just the second power $I(G)^2$. It is known that forbidding a cricket, a diamond, and a 4-cycle forces $I(G)^q$ to have a linear resolution for all $q\geq 2$. So, one of the results we will discuss is about a construction of gapfree graphs that do contain those subgraphs (and a 5-cycle), yet still have linear quotients for every $q\geq 2$. The talk will be example-driven, with quick primers on linear resolutions and linear quotients, and open questions at the end. Joint work with Erey, Faridi, Hà, Hibi, and Morey.