Higher discrete homotopy groups of graphs

Abstract

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 $n\ge 2$. Second we exhibit for each $n\ge 1$ 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 $n\ge 1$ there are graphs $G$ for which $A_n(G)$ is nontrivial.

21 pages, 10 figures