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.