Square-free graphs are multiplicative

Graph homomorphism is an ubiquitous notion in graph theory with a variety of applications, and a first step to understanding more general constraints given by relational structures, see e.g. the monograph of Hell and Nešetřil [5]. We write μ : G→ H if μ is a homomorphism from the graph G to H, or simply G → H if such a homomorphism exists. For example, a graph G is k-colorable iff it has a homomorphisms to the complete graph on k vertices, G→ Kk. The tensor product G×H is a natural operation arising in this context, defined as having a vertex for every pair (g, h) ∈ V (G)×V (H), and an edge between (g, h) and (g′, h′) whenever gg′ ∈ E(G) and hh′ ∈ E(H). It coincides with the so called categorical product in the category of graphs, with homomorphisms as arrows. In particular, for graphs K,G,H we have that K → G×H if and only if K → G and K → H. Hedetniemi [4] conjectured the following for the chromatic number of a product of graphs: