A degree condition for diameter two orientability of graphs

TL;DR

This paper determines the minimum degree threshold for graphs to be orientable with diameter two, establishing that it is approximately half the number of vertices plus a logarithmic term.

Contribution

It precisely characterizes the degree condition needed for diameter two orientability in graphs, refining previous bounds with an asymptotic formula.

Findings

The threshold degree _n is _n=rac{n}{2} + \u039b(\u03bb n).

Graphs with minimum degree above this threshold admit diameter two orientations.

The result sharpens understanding of graph orientation properties related to degree conditions.

Abstract

For $n \in \mathbb{N}$ let $\delta_n$ be the smallest value such that every graph of order $n$ and minimum degree at least $\delta_n$ admits an orientation of diameter two. We show that $\delta_n=\frac{n}{2} + \Theta(\ln n)$.