Shellability of simplicial complexes and simplicial complexes with the free vertex property

To a simplicial complex ∆, we associate a square-free monomial ideal F(∆) in the polynomial ring generated by its facet over a field. Furthermore, we could consider F(∆) as the Stanley–Reisner ideal of another simplicial complex δN (F(∆)) from facet ideal theory and Stanley–Reisner theory. In this paper, we determine what families of simplicial complexes ∆ have the property that their Stanley–Reisner complexes δN (F(∆)) are shellable. Furthermore, we show that the simplicial complex with the free vertex property is sequentially Cohen–Macaulay. This result gives a new proof for a result of Faridi on the sequentially Cohen–Macaulayness of simplicial forests.