Minimal Reductions and Cores of Edge Ideals

TL;DR

This paper investigates the structure of minimal reductions and cores of edge ideals of graphs, establishing key restrictions, equalities for specific graph classes, and counterexamples highlighting limitations of existing formulas.

Contribution

It provides new results on the containment and computation of cores of edge ideals, especially for even cycles, and identifies cases where known formulas do not apply.

Findings

Core equals intersection of minimal reductions for even cycles.

Core is contained in maximal ideal times the ideal when not basic.

Counterexample shows the core formula fails for some graphs.

Abstract

We study minimal reductions of edge ideals of graphs and determine restrictions on the coefficients of the generators of these minimal reductions. We prove that when $I$ is not basic, then $\core{I}\subset \m I$, where $I$ is an edge ideal in the corresponding localized polynomial ring and $\m$ is the maximal ideal of this ring. We show that the inclusion is an equality for the edge ideal of an even cycle with an arbitrary number of whiskers. Moreover, we show that the core is obtained as a finite intersection of homogeneous minimal reductions in the case of even cycles. The formula for the core does not hold in general for the edge ideal of any graph and we provide a counterexample. In particular, we show in this example that the core is not obtained as a finite intersection of general minimal reductions.