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HOMOLOGICAL LINEAR QUOTIENTS AND EDGE IDEALS OF GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 2 / October 2024
- Published online by Cambridge University Press:
- 29 January 2024, pp. 291-302
- Print publication:
- October 2024
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- Article
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It is well known that the edge ideal
$I(G)$ of a simple graph G has linear quotients if and only if 
$G^c$ is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph 
$G^c$ when 
$I(G)$ has homological linear quotients results in a graph with the same property. In particular, 
$I(G)$ has homological linear quotients when 
$G^c$ is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, 
$I(G)$ has homological linear quotients for every graph G such that 
$G^c$ is a 
$\lambda $-minimal chordal graph.
