  In this article, we consider the bipartite graphs $K_2 \times K_n$. We prove
that the connectedness of the complex $\displaystyle \text{Hom}(K_2\times
K_{n}, K_m) $ is $m-n-1$ if $m \geq n$ and $m-3$ in the other cases. Therefore,
we show that for this class of graphs, $\text{Hom} (G, K_m)$ is exactly $m-d-2$
connected, $m \geq n$, where $d$ is the maximal degree of the graph $G$.
