$r$-fundamental groups of graphs

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.