For every k≥2, a Cohen-Macaulay very well-covered whiskered bipartite graph has non-linear HS_k, and the paper identifies classes where HS_k does have linear quotients.
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On the homological shifts of cover ideals of Cohen-Macaulay graphs
For every k≥2, a Cohen-Macaulay very well-covered whiskered bipartite graph has non-linear HS_k, and the paper identifies classes where HS_k does have linear quotients.