  We show that every pair of longest paths in a $k$-connected graph on $n$
vertices intersect each other in at least $(8k-n+2)/5$ vertices. We also show
that, in a 4-connected graph, every pair of longest paths intersect each other
in at least four vertices. This confirms a conjecture of Hippchen for
$k$-connected graphs when $k\leq 4$ or $k\geq (n-2)/3$.
