  In this paper, we introduce the notions of $r$-fundamental groups of graphs,
$r$-covering maps, and $r$-neighborhood complexes of graphs for a positive
integer $r$. There is a natural correspondence between $r$-covering maps and
$r$-fundamental groups as is the case of the covering space theory in topology.
We can derive obstructions of the existences of graph maps from $r$-fundamental
groups. Especially, $r$-fundamental groups gives deep informations about the
existences of graph maps to odd cycles. For example, we prove the Kneser graph
$K_{2k+1,k}$ has no graph maps to $C_5$. $r$-neighborhood complexes are natural
generalization of neighborhood complexes defined by Lov$\acute{\rm a}$sz. We
prove that $(2r)$-fundamental groups gives graph theoretical description of the
fundamental groups of $r$-neighborhood complexes.
