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

TL;DR

This paper explores a graph operation inverse to taking graph powers, showing it preserves topological properties and relates graph homomorphisms to topological maps, with implications for Hedetniemi's conjecture and topological conjectures.

Contribution

It introduces a natural inverse graph operation that preserves the topology of the box complex and links graph homomorphisms to topological maps, connecting combinatorics and topology.

Findings

The inverse operation preserves the $bZ_2$-homotopy type of the box complex.

Existence of a $bZ_2$-map between box complexes corresponds to a homomorphism after applying the inverse operation.

Implications for Hedetniemi's conjecture suggest a topological analogue involving $bZ_2$-spaces.

Abstract

We consider a natural graph operation $\Omega_k$ 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 $\mathbb{Z}_2$-homotopy type) of the box complex, a basic tool in topological combinatorics. Moreover, we prove that the box complex of a graph G admits a $\mathbb{Z}_2$-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph $\Omega_k(G)$ 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 were true for n-colorings, then the following analogous conjecture in topology would also also true: If X,Y are $\mathbb{Z}_2$-spaces (finite $\mathbb{Z}_2$-simplicial complexes) such that X x Y admits a $\mathbb{Z}_2$-map to the (n-2)-dimensional sphere, then X or Y itself admits such a map. We discuss this and other implications, arguing the importance of the topological conjecture.