Balanced Vertex Decomposable Simplicial Complexes and their h-vectors

Given any finite simplicial complex $\Delta$, we show how to construct from a colouring $\chi$ of $\Delta$ a new simplicial complex $\Delta_{\chi}$ that is balanced and vertex decomposable. In addition, the $h$-vector of $\Delta_{\chi}$ is precisely the $f$-vector of $\Delta$.  Our construction generalizes the "whiskering'' construction of Villarreal, and Cook and Nagel. We also reverse this construction to prove a special case of a conjecture of Cook and Nagel, and Constantinescu and Varbaro on the $h$-vectors of flag complexes.