Cohen–Macaulay, shellable and unmixed clutters with a perfect matching of König type

Let CC be a clutter with a perfect matching e1,…,ege1,…,eg of König type and let ΔCΔC be the Stanley–Reisner complex of the edge ideal of CC. If all c-minors of CC have a free vertex and CC is unmixed, we show that ΔCΔC is pure shellable. We are able to describe, in combinatorial and algebraic terms, when ΔCΔC is pure. If CC has no cycles of length 3 or 4, then it is shown that ΔCΔC is pure if and only if ΔCΔC is pure shellable (in this case eiei has a free vertex for all ii), and that ΔCΔC is pure if and only if for any two edges f1,f2f1,f2 of CC and for any eiei, one has that f1∩ei⊂f2∩eif1∩ei⊂f2∩ei or f2∩ei⊂f1∩eif2∩ei⊂f1∩ei. It is also shown that this ordering condition implies that ΔCΔC is pure shellable, without any assumption on the cycles of CC. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen–Macaulay, and they have linear resolutions. Furthermore if CC is admissible and complete, then CC is unmixed. We characterize certain conditions that occur in a Cohen–Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi–on the structure of unmixed simplicial trees–to clutters with the König property without 3-cycles or 4-cycles.