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 $t=(n-1)/2$.