(Almost) Complete Intersection Lovász-Saks-Schrijver ideals and regularity of their powers
Marie Amalore Nambi, Neeraj Kumar, Chitra Venugopal
TL;DR
We study when Lovász–Saks–Schrijver (LSS) ideals and twisted LSS-ideals associated with graphs are (almost) complete intersections and how these properties control the regularity of their powers. A key tool is the twisted positive matching decomposition $\mathrm{tpmd}(G)$; we prove that $d \ge \mathrm{tpmd}(G)$ implies $\hat{L}_G(d)$ is a radical complete intersection, with connections to determinantal and Pfaffian ideals for unicyclic and bicyclic graphs. We characterize almost complete intersection LSS-ideals for trees and $C_3$-free unicyclic/bicyclic graphs, including explicit constructions and height calculations, and provide exact formulas and bounds for the regularity of powers in trees and unicyclic graphs with $\Delta(G) \le d$, along with Koszulness criteria for quotients $S/L_G(d)$. These results translate combinatorial graph invariants into precise algebraic properties, yielding concrete criteria and bounds for the ideals and their powers across several graph families.
Abstract
We discuss the property of (almost) complete intersection of LSS-ideals of graphs of some special forms, like trees, unicyclic, and bicyclic graphs. Further, we give a sufficient condition for the complete intersection property of twisted LSS-ideals in terms of a new graph theoretical invariant called twisted positive matching decomposition number denoted by tpmd. We also study the regularity of powers of LSS-ideals and make observations related to the Koszul property of the quotients of the same.