Hom complexes of graphs of diameter $1$

Abstract: Given a finite simplicial complex $X$ and a connected graph $T$ of diameter $1$, in \cite{anton} Dochtermann had conjectured that $\text{Hom}(T,G_{1,X})$ is homotopy equivalent to $X$. Here, $G_{1, X}$ is the reflexive graph obtained by taking the $1$-skeleton of the first barycentric subdivision of $X$ and adding a loop at each vertex. This was proved by Dochtermann and Schultz in \cite{ds12}. In this article, we give an alternate proof of this result by understanding the structure of the cells of Hom$(K_n,G_{1,X})$, where $K_n$ is the complete graph on $n$ vertices. We prove that the neighborhood complex of $G_{1,X}$ is homotopy equivalent to $X$ and Hom$(K_n,G_{1,X})\simeq $ Hom$(K_{n-1},G_{1,X})$, for each $n\geq 3$.