  Borodin and Kostochka conjectured that every graph $G$ with maximum degree
$\Delta \ge 9$ satisfies $\chi \le \max\{\omega, \Delta-1\}$. We carry out an
in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture.
Our main tool is the identification of graph joins that are $f$-choosable,
where $f(v) = d(v) - 1$ for each vertex $v$. Since such a join cannot be an
induced subgraph of a vertex critical graph with $\chi = \Delta$, we have a
wealth of structural information about minimum counterexamples to the
Borodin-Kostochka conjecture.
  Our main result proves that certain conjectures that are prima facie weaker
than the Borodin-Kostochka conjecture are in fact equivalent to it. One such
equivalent conjecture is the following: Any graph with $\chi \ge \Delta = 9$
contains $K_3 * \bar{K_6}$ as a subgraph.
