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

Let image be a clutter with a perfect matching e1,…,eg of König type and let image be the Stanley–Reisner complex of the edge ideal of image. If all c-minors of image have a free vertex and image is unmixed, we show that image is pure shellable. We are able to describe, in combinatorial and algebraic terms, when image is pure. If image has no cycles of length 3 or 4, then it is shown that image is pure if and only if image is pure shellable (in this case ei has a free vertex for all i), and that image is pure if and only if for any two edges f1,f2 of image and for any ei, one has that f1∩eisubset off2∩ei or f2∩eisubset off1∩ei. It is also shown that this ordering condition implies that image is pure shellable, without any assumption on the cycles of image. 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 image is admissible and complete, then image 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.