A monomial ideal I is said to have homological linear quotients if, for each \(k\ge 0\) , the homological shift ideal \({{\,\textrm{HS}\,}}_k(I)\) 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 \(\{{{\,\textrm{H}\,}}_n^c\}_{n\ge 6}\) and propose a conjecture that an edge ideal I(G) has homological linear quotients if and only if G is co-chordal and \({{\,\textrm{H}\,}}_n^c\) -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 \({{\,\textrm{HS}\,}}_k(I(G))\) has linear quotients.
