Linearity defect of edge ideals and Fr\"oberg's theorem

TL;DR

This paper extends Fr"oberg's theorem by classifying graphs with edge ideals of linearity defect at most 1, revealing they are exactly weakly chordal graphs with induced matching number at most 2, independent of the field characteristic.

Contribution

The authors classify graphs with edge ideals of linearity defect ≤ 1, generalizing Fr"oberg's theorem and providing explicit characterizations involving weakly chordal graphs.

Findings

Classified graphs with linearity defect ≤ 1 as weakly chordal with induced matching number ≤ 2

Computed linearity defect for cycles and weakly chordal graphs

Recovered and extended results on projective dimension and regularity of edge ideals

Abstract

Fr\"oberg's classical theorem about edge ideals with $2$-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most $1$. Our characterization is independent of the characteristic of the base field: the graphs in question are exactly weakly chordal graphs with induced matching number at most $2$. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, H\`a, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann-Engstr\"om, Kimura, and Woodroofe on the projective dimension and Castelnuovo-Mumford regularity of edge ideals.