Higher discrete homotopy groups of graphs
Bob Lutz · Algebraic Combinatorics · 2021
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 ≥ 2 . Second we exhibit for each n ≥ 1 a natural homomorphism ψ : A n ( G ) → ℋ n ( G ) , where ℋ n ( G ) is the n th discrete cubical singular homology group, and an infinite family of graphs G for which ℋ n ( G ) is nontrivial and ψ is surjective. It follows that for each n ≥ 1 there are graphs G for which A n ( G ) is nontrivial.