Connectedness of Certain Graph Coloring Complexes

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 $G$.