  After long term efforts, it was recently proved in \cite{DKM2} that except
for the Peterson graph, every connected vertex-transitive graph of order $rs$
has a Hamilton cycle, where $r$ and $s$ are primes. A natural topic is to solve
the hamiltonian problem for connected vertex-transitive graphs of $2rs$. This
topic is quite trivial, as the problem is still unsolved even for that of
$r=3$. In this paper, it is shown that except for the Coxeter graph, every
connected vertex-transitive graph of order $2rs$ contains a Hamilton cycle,
provided the automorphism group acts primitively on vertices.
