Higher discrete homotopy groups of graphs

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 graph containing no 3- or 4-cycles, then the nth group A n (G) trivial all n?2. Second exhibit each n?1 natural homomorphism ?:A (G)?? (G), where ? cubical singular homology group, and an infinite family of which nontrivial ? surjective. It follows there are nontrivial.