On a Vizing-like conjecture for direct product graphs

Let γ(G) be the domination number of a graph G, and let G × H be the direct product of graphs G and H. It is shown that for any k ⩾ 0 there exists a graph G such that γ(G × G) ⩽ γ(G)2 − k. This in particular disproves a conjecture from [5].