Edge ideals with linear quotients and without homological linear quotients

Paper

Abstract

A monomial ideal

is said to have homological linear quotients if for each

, the homological shift ideal

has linear quotients. It is a well-known fact that if an edge ideal

has homological linear quotients, then

is co-chordal. We construct a family of co-chordal graphs

and propose a conjecture that an edge ideal

has homological linear quotients if and only if

is co-chordal and

-free for any

. In this paper, we prove one direction of the conjecture. Moreover, we study possible patterns of pairs

of a co-chordal graph

and integer

such that

has linear quotients.