On Link-irregular Digraphs
Alexander Bastien, Omid Khormali
TL;DR
The paper extends link-irregularity to digraphs by defining directed links and establishing existence thresholds and structural constraints. It proves that link-irregular digraphs exist iff $n\ge5$ and that their underlying graphs contain a $3$-cycle, and it investigates tournaments with this property, providing constructions for small $n$ and computational verification up to $n\le100$. It also derives degree and labeling bounds, shows most link-irregular digraphs are nonplanar, and presents explicit dense and regular examples, including a link-irregular regular tournament on 9 vertices, plus relations to link-irregular labeling. The work raises conjectures about link-irregular tournaments for all $n\ge6$ and offers algorithmic approaches to identify such digraphs.
Abstract
We extend the study of link-irregular graphs to directed graphs (digraphs), where a digraph is link-irregular if no two vertices have isomorphic directed links. We establish that link-irregular digraphs exist on $n$ vertices if and only if $n \geq 5$, and prove that their underlying graphs must contain 3-cycles. We conjecture that link-irregular tournaments exist if and only if $n \geq 6$, providing explicit constructions for $n \leq 8$ and computational verification for $n \leq 100$. We derive lower bounds on the minimum degree and outdegree required for link-irregularity, establish that almost all link-irregular digraphs are nonplanar, and prove that any link-irregular orientable graph admits a link-irregular labeling. Additionally, we construct explicit examples of link-irregular digraphs with constant outdegree and regular tournaments.