An improved upper bound on the oriented diameter of graphs with diameter $4$

TL;DR

This paper improves the upper bound on the oriented diameter of bridgeless graphs with diameter 4 from 21 to 18, advancing understanding of graph orientations.

Contribution

It provides a tighter upper bound on the oriented diameter for graphs with diameter 4, refining previous bounds.

Findings

Improved upper bound of f(4) to 18

Confirmed f(3)=9 for graphs with diameter 3

Reviewed previous bounds for f(d) in literature

Abstract

Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chv\'atal and Thomassen (1978) established general bounds for $f(d)$ which implies $f(4)\geq 12$, and proved $f(2)=6$. Kwok et al. (2010) proved that $9\leq f(3)\leq 11$. Wang and Chen (2022) determined $f(3)=9$. Babu et al. (2021) showed $f(4)\leq 21$. In this paper, we improve the upper bound of $ f(4) $ to $18$.