Note on the oriented diameter of graphs with diameter 3

TL;DR

This paper investigates the minimum oriented diameter of bridgeless graphs with diameter 3, refining bounds and identifying the smallest possible diameter in such graphs with minimal edges.

Contribution

It determines the exact oriented diameter for bridgeless graphs with diameter 3 that have the fewest edges, improving previous bounds.

Findings

Refined bounds on the oriented diameter for diameter 3 graphs.

Identified the minimal edge count for such graphs.

Established the exact oriented diameter in the minimal edge case.

Abstract

In 1978, Chv\'atal and Thomassen showed that every bridgeless graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value $f(d)$ such that every bridgeless graph $G$ with diameter $d$ has an orientation with diameter at most $f(d)$. For $d=3$, they proved that $8\leq f(d)\leq 24$. Until recently, Kwok, Liu and West improved the above bounds by proving $9\leq f(3)\leq 11$ in [P.K. Kwok, Q. Liu and D.B. West, Oriented diameter of graphs with diameter 3, J. Combin. Theory Ser.B 100(2010), 265-274]. In this paper, we determine the oriented diameter among the bridgeless graphs with diameter 3 that have minimum number of edges.