On the oriented diameter of graphs with given minimum degree

TL;DR

This paper establishes that for all graphs with minimum degree at least 2, the oriented diameter is at most the same order as the undirected diameter bound, confirming the optimal constant factor of 1.

Contribution

It proves that the smallest universal constant c for the bound on oriented diameter relative to the undirected diameter is exactly 1, improving previous bounds.

Findings

The optimal constant c in the bound is 1.

The bound applies to all graphs with minimum degree at least 2.

Previous bounds had larger constants, now improved to the best possible.

Abstract

Erd\H{o}s, Pach, Pollack, and Tuza [\textit{J. Combin. Theory Ser. B, 47(1) (1989), 73-79}] proved that the diameter of a connected $n$-vertex graph with minimum degree $\delta$ is at most $\frac{3n}{\delta+1}+O(1)$. The oriented diameter of an undirected graph $G$, denoted by $\overrightarrow{\text{diam}}(G)$, is the minimum diameter of a strongly connected orientation of $G$. Bau and Dankelmann [\textit{European J. Combin., 49 (2015), 126-133}] showed that for every bridgeless $n$-vertex graph $G$ with minimum degree $\delta$, $\overrightarrow{\text{diam}}(G) \leq \frac{11n}{\delta+1}+9$. They also showed an infinite family of graphs with oriented diameter at least $\frac{3n}{\delta+1} + O(1)$ and posed the problem of determining the smallest possible value $c$ for which $\overrightarrow{\text{diam}}(G) \leq c \cdot\frac{3n}{\delta+1}+O(1)$ holds. In this paper, we show that the smallest value $c$ such that the upper bound above holds for all $\delta\geq 2$ is $1$, which is best possible.