On inverse powers of graphs and topological implications of Hedetniemi's conjecture

Abstract

We consider a natural graph operation Ωk that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the Z2-homotopy type) of the box complex, a basic tool in applications of topology in combinatorics. Moreover, we prove that the box complex of a graph G admits a Z2-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph Ωk(G) admits a homomorphism to H, for high enough k.This allows to show that if Hedetniemi's conjecture on the chromatic number of graph products is true, then the following analogous conjecture in topology is also true: If n∈N and X,Y are Z2-spaces (finite Z2-simplicial complexes) such that X×Y admits a Z2-map to the n-dimensional sphere, then X or Y itself admits such a map. We discuss this and other implications, arguing the importance of the topological conjecture.