The diameter of strong orientations of strong products of graphs

Abstract

Let $G$ and $H$ be graphs, and $G\boxtimes H$ 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 $G\boxtimes H$ such that ${\rm diam}(D)\leq 2r+15$, where $r$ is the radius of $G\boxtimes H$.This improves the general bound ${\rm diam}(D)\leq 2r^2+2r$ for arbitrary graphs, proved by Chv${\rm \acute{a}}$tal and Thomassen.