  A connected $n$-chromatic graph $G$ is double-critical if for all the edges
$xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and
Lov\'asz conjectured that the only double-critical $n$-chromatic graph is
$K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we
verify this conjecture for claw-free graphs $G$ of chromatic number $6$.
