A counterexample to a conjecture of Bj\"{o}rner and Lov\'asz on the $\chi$-coloring complex

Associated with every graph $G$ of chromatic number $\chi$ is another graph $G^{\mathrm{\prime }}$. The vertex set of $G^{\mathrm{\prime }}$ consists of all $\chi$-colorings of $G$, and two $\chi$-colorings are adjacent when they differ on exactly one vertex. According to a conjecture of Bj"{o}rner and Lov'asz, this graph $G^{\mathrm{\prime }}$ must be disconnected. In this note we give a counterexample to this conjecture.