On the index of powers of edge ideals

TL;DR

This paper investigates how the index of powers of edge ideals behaves, showing it increases for ideals with linear relations but not necessarily for higher-degree monomial ideals.

Contribution

It provides new insights into the behavior of the index of powers of edge ideals, especially highlighting the difference based on the degree of generators.

Findings

Index increases with powers for ideals with linear relations.

Counterexamples show non-monotonic behavior for higher-degree monomial ideals.

Results deepen understanding of minimal free resolutions of edge ideals.

Abstract

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal $I$ is strictly increasing if $I$ has linear relations. Examples show that this need not to be the case for monomial ideals generated in degree greater than two.