  Let $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite
field with $q$ elements. Construct a graph, called the distance graph of $E$,
by letting the vertices be the elements of $E$ and connect a pair of vertices
corresponding to vectors $x,y \in E$ by an edge if
$||x-y||={(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2=1$. We shall prove that if the size
of $E$ is sufficiently large, then the distance graph of $E$ contains long
non-overlapping paths and vertices of high degree.
