Multiplicity Bounds for Quadratic Monomial Ideals

TL;DR

This paper proves specific multiplicity bounds for quadratic monomial ideals, focusing on bipartite graphs and their algebraic properties, and characterizes cases of equality and Cohen-Macaulay conditions.

Contribution

It establishes the strong and Taylor bound conjectures for quadratic monomial ideals, linking graph operations to algebraic properties and resolutions.

Findings

Proved the strong conjecture for bipartite graph edge ideals.

Verified the Taylor bound conjecture for all quadratic monomial ideals.

Characterized bipartite graphs with Cohen-Macaulay edge ideals.

Abstract

We prove the multiplicity bounds conjectured by Herzog-Huneke-Srinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of bipartite graphs, and the weaker Taylor bound conjecture for all quadratic monomial ideals. We attach a directed graph to a bipartite graph with perfect matching, and describe operations on the directed graph that would reduce the problem to a Cohen-Macaulay bipartite graph. We determine when equality holds in the conjectured bound for edge ideals of bipartite graphs, and verify that when equality holds, the resolution is pure. We characterize bipartite graphs that have Cohen-Macaulay edge ideals and quasi-pure resolutions.