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

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 bound given in (\ref{abstract}) is sharp. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy. We were also able to find a class of exponential graphs, which are homotopy test graphs. Hedetniemi's conjecture states that the chromatic number of the categorical product 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 enrich the family of graphs satisfying the Hedetniemi's conjecture.