Large Girth and Small Oriented Diameter Graphs

TL;DR

This paper improves bounds on the oriented diameter of graphs by incorporating girth, showing that larger girth can lead to smaller diameters in graph orientations, refining previous results.

Contribution

It introduces a new bound involving girth and degree, providing a tighter estimate for the oriented diameter of graphs, especially for graphs with girth three.

Findings

Established a bound of the form $(2g+ ext{small } ext{epsilon})rac{n}{h( ext{degree},g)}+O(1)$

Improved previous bounds for graphs with girth three

Demonstrated the impact of girth on the oriented diameter

Abstract

In 2015, Dankelmann and Bau proved that for every bridgeless graph $G$ of order $n$ and minimum degree $\delta$ there is an orientation of diameter at most $11\frac{n}{\delta+1}+9$. In 2016, Surmacs reduced this bound to $7\frac{n}{\delta+1}.$ In this paper, we consider the girth of a graph $g$ and show that for any $\varepsilon>0$ there is a bound of the form $(2g+\varepsilon)\frac{n}{h(\delta,g)}+O(1)$, where $h(\delta,g)$ is a polynomial. Letting $g=3$ and $\varepsilon<1$ gives an inprovement on the result by Surmacs.