Neighborhood Complexes of Kneser Graphs, $KG_{3,k}$

In this article, we prove that the neighborhood complex of the Kneser graph $KG_{3,k}$ is of the same homotopy type as that of a wedge of $\frac{(k+1)(k+3)(k+4)(k+6)}{4}+1$ spheres of dimension $k$. We construct a maximal subgraph $S_{3,k}$ of $KG_{3,k}$, whose neighborhood complex deformation retracts onto the neighborhood complex of $SG_{3,k}$.