  The neighborhood complexes of graphs were introduced by Lov\'asz in his proof
of the Kneser conjecture. He showed that a certain topological property of
$N(G)$ gives a lower bound for the chromatic number of $G$.
  In this paper, we study a combinatorial description of the fundamental groups
of the neighborhood complexes. For a positive integer $r$, we introduce the
$r$-fundamental group $\pi_1^r(G,v)$ of a based graph $(G,v)$ and the
$r$-neighborhood complex $N_r(G)$ of $G$. The $1$-neighborhood complex is the
neighborhood complex. We show that the even part $\pi_1^{2r}(G,v)_{ev}$, which
is a subgroup of $\pi_1^{2r}(G,v)$ with index 1 or 2, is isomorphic to the
fundamental group of $(N_r(G),v)$ if $v$ is not isolated. We can use the
$r$-fundamental groups to show the non-existence of graph homomorphisms. For
example, we show that $\pi_1^3(KG_{2k+1,k})$ is isomorphic to $\mathbb{Z} /2$,
and this implies that there is no graph homomorphism from $KG_{2k+1,k}$ to the
5-cycle graph $C_5$. We discuss the covering maps associated to $r$-fundamental
groups.
