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The neighborhood complex $\N(G)$ of a graph $G$ were introduced by L.Lov{\'a}sz in his proof of Kneser conjecture. He proved that for any graph $G$,  \begin{align} \label{abstract} \chi(G) \geq conn(\N(G))+3.  \end{align}  In this article we show that for a class of exponential graphs the boundgiven in (\ref{abstract}) is sharp. Further, we show that the neighborhoodcomplexes of these exponential graphs are spheres up to homotopy.  We were also able to find a class of exponential graphs, which are homotopytest graphs.  Hedetniemi's conjecture states that the chromatic number of the categoricalproduct of two graphs is the minimum of the chromatic number of the factors.Let $M(G)$ denotes the Mycielskian of a graph $G$. We show that, for any graph$G$ containing $M(M(K_n))$ as a subgraph and for any graph $H$, if $\chi(G\times H) = n+1$, then $\min\{\chi(G), \chi(H)\} = n+1$. Therefore, we enrichthe family of graphs satisfying the Hedetniemi's conjecture.

Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi