Square-free graphs are multiplicative

Abstract

A graph K is square-free if it contains no four-cycle as a subgraph (i.e., for every quadruple of vertices, if v0v1,v1v2,v2v3,v3v0∈E(K), then v0=v2 or v1=v3). A graph K is multiplicative if G×H→K implies G→K or H→K, for all graphs G, H. Here G×H is the tensor (or categorical) graph product and G→K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture, which states that χ(G×H)=min⁡(χ(G),χ(H)), is equivalent to the statement that all cliques Kn are multiplicative. However, the only non-trivial graphs known to be multiplicative are K3, odd cycles, and still more generally, circular cliques Kp/q with 2≤pq<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 with 2≤pq<4.