A Size Condition for Diameter Two Orientable Graphs

TL;DR

This paper proves a conjecture that large enough graphs with a certain number of edges always have an orientation with diameter two, determining the minimal edge count for this property.

Contribution

It confirms Koh and Tay's conjecture, establishing the exact minimum size for graphs to admit a diameter-two orientation for all n ≥ 5.

Findings

Confirmed the conjecture for all n ≥ 5

Determined the minimal edge count for diameter-two orientations

Extended understanding of graph orientations with diameter constraints

Abstract

It was conjectured by Koh and Tay [Graphs Combin. 18(4) (2002), 745--756] that for $n\geq 5$ every simple graph of order $n$ and size at least $\binom{n}{2}-n+5$ has an orientation of diameter two. We prove this conjecture and hence determine for every $n\geq 5$ the minimum value of $m$ such that every graph of order $n$ and size $m$ has an orientation of diameter two.