The diameter of strong orientations of strong products of graphs

TL;DR

This paper establishes an improved upper bound on the diameter of strongly connected orientations of strong graph products, reducing previous bounds significantly for connected graphs.

Contribution

It proves a tighter bound on the diameter of strong orientations of strong graph products, improving upon earlier general bounds for connected graphs.

Findings

Diameter of strong orientations is at most 2r+15 for connected graphs.

Improves previous bound of 2r^2+2r for arbitrary graphs.

Results apply specifically to strong products of connected 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.