  This paper studies a discrete homotopy theory for graphs introduced by
Barcelo et al. We prove two main results. First we show that if $G$ is a graph
containing no 3- or 4-cycles, then the $n$th discrete homotopy group $A_n(G)$
is trivial for all $n\geq 2$. Second we exhibit for each $n\geq 1$ a natural
homomorphism $\psi:A_n(G)\to \mathcal{H}_n(G)$, where $\mathcal{H}_n(G)$ is the
$n$th discrete cubical singular homology group, and an infinite family of
graphs $G$ for which $\mathcal{H}_n(G)$ is nontrivial and $\psi$ is surjective.
It follows that for each $n\geq 1$ there are graphs $G$ for which $A_n(G)$ is
nontrivial.
