Higher discrete homotopy groups of graphs

TL;DR

This paper investigates discrete homotopy groups of graphs, proving their triviality under certain conditions and establishing a connection with discrete homology groups, revealing nontrivial cases.

Contribution

It demonstrates conditions for triviality of discrete homotopy groups and constructs a natural homomorphism linking these groups to discrete homology, showing nontrivial examples.

Findings

Graphs with no 3- or 4-cycles have trivial higher homotopy groups.

A natural homomorphism from homotopy to homology groups is constructed.

Existence of graphs with nontrivial homology and nontrivial homotopy groups.

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 $n\geq 2$. 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.