Edge ideals with linear quotients and without homological linear quotients

A monomial ideal $I$ is said to have homological linear quotients if for each $k\ge 0$, the homological shift ideal ${\mathrm{H}\mathrm{S}}_k(I)$ has linear quotients. It is a well-known fact that if an edge ideal $I(G)$ has homological linear quotients, then $G$ is co-chordal. We construct a family of co-chordal graphs $\{{\mathrm{H}}_n^c{\}}_{n\ge 6}$ and propose a conjecture that an edge ideal $I(G)$ has homological linear quotients if and only if $G$ is co-chordal and ${\mathrm{H}}_n^c$-free for any $n\ge 6$. In this paper, we prove one direction of the conjecture. Moreover, we study possible patterns of pairs $(G,k)$ of a co-chordal graph $G$ and integer $k$ such that ${\mathrm{H}\mathrm{S}}_k(I(G))$ has linear quotients.