Square-free graphs are multiplicative

TL;DR

This paper proves that all square-free graphs are multiplicative, expanding the class of known multiplicative graphs beyond simple cases like odd cycles and circular cliques, and provides new topological insights.

Contribution

It establishes that all square-free graphs are multiplicative, offering the first such results for graphs with chromatic number greater than 4 and providing a new proof for circular cliques.

Findings

All square-free graphs are multiplicative.

First multiplicative graphs with chromatic number > 4.

New topological proof for circular cliques.

Abstract

A graph K is square-free if it contains no four-cycle as a subgraph. A graph K is multiplicative if GxH -> K implies G -> K or H -> K, for all graphs G,H. Here GxH is the tensor (or categorical) graph product and G -> K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture states that all cliques K_n are multiplicative. However, the only non-trivial graphs known to be multiplicative are K_3, odd cycles, and still more generally, circular cliques $K_{p/q}$ with 2 <= p/q < 4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In particular, this gives the first multiplicative graphs of chromatic number higher than 4. Generalizing, in terms of the box complex, the topological insight behind existing proofs for odd cycles, we also give a different proof for circular cliques.