The diameter of strong orientations of strong products of graphs

Let $G$ and $H$ be graphs, and $G⊠H$ the strong product of $G$ and $H$. We prove that for any connected graphs $G$ and $H$ there is a strongly connected orientation $D$ of $G⊠H$ such that ${\rm diam}(D)\leq 2r+15$,where$r$istheradiusof$G\boxtimes H$.This improves the general bound $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(D)\le 2r^2+2r$ for arbitrary graphs, proved by Chv${\rm \acute{a}}$tal and Thomassen.