Lov\'asz-Saks-Schrijver ideals and parity binomial edge ideals of graphs

TL;DR

This paper classifies graphs based on when their Lovász-Saks-Schrijver and parity binomial edge ideals are complete or almost complete intersections, and explores algebraic properties like Cohen-Macaulayness and Betti numbers.

Contribution

It provides a complete classification of graphs with these ideal properties and analyzes their algebraic structures, including Rees and symmetric algebras.

Findings

Classified graphs with complete intersection LSS and parity binomial edge ideals.

Proved Cohen-Macaulayness of Rees algebra for almost complete intersections.

Computed Betti numbers and described minimal presentations for specific graph classes.

Abstract

Let $G$ be a simple graph on $n$ vertices. Let $L_G \text{ and } \mathcal{I}_G \: $ denote the Lov\'asz-Saks-Schrijver(LSS) ideal and parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n] $ respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we prove that their Rees algebra is Cohen-Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS ideals of trees and odd unicyclic graphs. We also obtain an explicit description of the defining ideal of the symmetric algebra of LSS ideals of trees and odd unicyclic graphs.