Vertex Decomposability of 2-CM and Gorenstein Simplicial Complexes of Codimension 3

Abstract

Let \(\Delta \) be a simplicial complex on vertex set \([n]\). It is shown that if \(\Delta \) is complete intersection, Cohen–Macaulay of codimension 2, Gorenstein of codimension 3, or 2-Cohen–Macaulay of codimension 3, then \(\Delta \) is vertex decomposable. As a consequence, we show that if \(\Delta \) is a simplicial complex such that \(I_\Delta = I_t(C_n)\), where \(I_t(C_n)\) is the path ideal of length \(t\) of \(C_n\), then \(\Delta \) is vertex decomposable if and only if \(t=n, t=n-1\), or \(n\) is odd and \(t=(n-1)/2\).