Oriented diameter of graphs with given domination number

TL;DR

This paper proves a conjecture that the oriented diameter of a connected bridgeless graph with domination number  is at most .5, and establishes a tight bound of 7-1 using induction and recursive structure.

Contribution

It confirms the conjectured bound on the oriented diameter based on domination number and introduces a recursive approach for future applications.

Findings

Confirmed the conjecture .5 bound is sharp.

Established a tight bound of 7-1 for the oriented strong diameter.

Introduced a recursive method using unavoidable subgraphs.

Abstract

Let $G$ be a connected bridgeless graph with domination number $\gamma$. The oriented diameter (strong diameter) of $G$ is the smallest integer $d$ for which $G$ admits a strong orientation with diameter (strong diameter) $d$. Kurz and L\"atsch (2012) conjectured the oriented diameter of $G$ is at most $\lceil \frac{7\gamma+1}{2}\rceil$ and the bound is sharp. In this paper, we confirm the conjecture by induction on $\gamma$ through contracting an unavoidable alternative subgraph, which holds potential for future applications. Moreover, we show the oriented strong diameter of $G$ is at most $7\gamma-1$ by using the same recursive structure, and the bound is best possible.