A Size Condition for Small Diameter Orientable Graphs

TL;DR

This paper investigates the minimum size of bridgeless graphs needed to ensure orientations with small diameter, proving a specific conjecture case and establishing bounds for others.

Contribution

It proves the conjecture for diameter $d=n-2$ and provides lower bounds for diameters between 5 and $n-2$, advancing understanding of graph orientation diameters.

Findings

Proved the conjecture for $d=n-2$ case.

Established lower bounds for $5 \\leq d \\leq n-2$.

Connected size conditions to orientation diameters in bridgeless graphs.

Abstract

In 2002, Koh and Tay conjectured that every bridgeless graph of order $n\geq 5$ and size at least ${n\choose 2}-n+5$ has an orientation of diameter two. Later, Cochran, Czabarka, Dankelmann and Sz\'{e}kely proved this conjecture and asked what is the minimum number of edges required in a bridgeless graph of order $n$ to guarantee the existence of an orientation of diameter at most $d$? We conjecture that the answer is ${n-d \choose 2}+n+2$. We prove this conjecture for the case $d=n-2$ and prove the lower bound of this conjecture for the case $5\leq d\leq n-2$.