A Coxeter group
$W$
is said to be rigid if, given any two Coxeter systems
$(W, S)$
and
$(W, S^\prime)$
, there is an automorphism
$\rho: W \longrightarrow W$
such that
$\rho(S) = S^\prime$
. The class of Coxeter systems
$(W, S)$
for which the Coxeter graph
$\Gamma_S$
is complete and has only odd edge labels is considered. (Such a system is said to be of type
$K_n$
.) It is shown that if
$W$
has a type
$K_n$
system, then any other system for
$W$
is also type
$K_n$
. Moreover, the multiset of edge labels on
$\Gamma_S$
and
$\Gamma_{S^\prime}$
agree. In particular, if all but one of the edge labels of
$\Gamma_S$
are identical, then
$W$
is rigid.