  A connected $k$-chromatic graph $G$ is double-critical if for all edges $uv$
of $G$ the graph $G - u - v$ is $(k-2)$-colourable. The only known
double-critical $k$-chromatic graph is the complete $k$-graph $K_k$. The
conjecture that there are no other double-critical graphs is a special case of
a conjecture from 1966, due to Erd\H{o}s and Lov\'asz. The conjecture has been
verified for $k \leq 5$. We prove for $k=6$ and $k=7$ that any non-complete
double-critical $k$-chromatic graph is 6-connected and has $K_k$ as a minor.
