On the oriented diameter of graphs with given minimum degree

TL;DR

The paper addresses bounding the oriented diameter of bridgeless graphs in terms of order $n$ and minimum degree $\delta$. It introduces a Main Lemma that iteratively builds bridgeless oriented subgraphs $H_i$ and vertex sets $S_i$ with diameter bounded by $(3+\epsilon)|S_i|$ and a neighborhood-expansion property, while ensuring all vertices are within distance $L(\epsilon)$ of the final $H_m$. This framework, combined with an extension lemma, yields an overall bound $\overrightarrow{\text{diam}}(G)\le (3+\epsilon)\frac{n}{\delta-2}+O(1)$ for all $\delta\ge 3$, and asymptotically tightness is discussed via constructions approaching the lower bound $\frac{3n}{\delta+1}+O(1)$. The results asymptotically resolve the question posed by Bau and Dankelmann on the smallest possible constant $c$ in $\overrightarrow{\text{diam}}(G)\le c\cdot\frac{3n}{\delta+1}+O(1)$. The work also notes potential improvements to the denominator and connections to algorithmic orientation construction.

Abstract

Erdős, 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 $δ$ is at most $\frac{3n}{δ+1}+O(1)$. 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 $δ$, $\overrightarrow{\text{diam}}(G) \leq \frac{11n}{δ+1}+9$. They also showed an infinite family of graphs with oriented diameter at least $\frac{3n}{δ+1} + O(1)$ and posed the problem of determining the smallest possible value $c$ for which $\overrightarrow{\text{diam}}(G) \leq c \cdot\frac{3n}{δ+1}+O(1)$ holds. In this paper, we show that the smallest value $c$ such that the upper bound above holds for all $δ\geq 2$ is $1$, which is best possible.

Figures (5)

• Figure 1: The main paths $P$ and $Q$.
• Figure 2: Independent paths between $a_1, a_2$ and $b_1$, and between $b_1, b_2$ and $a_1$.
• Figure 3: Illustrations for Case $2$--$4$ (from left to right).
• Figure 5: Cases when $x'\in B_i"$.
• Figure 6: Cases when $x'\in B_i'\backslash A_i'$.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1
• Lemma 2.1
• proof : Proof of Lemma \ref{['lem:main-lemma']}
• Claim 1
• proof
• Claim 2
• proof
• Claim 3