On Link-irregular Digraphs

TL;DR

This paper extends the concept of link-irregular graphs to directed graphs, establishing existence conditions, structural properties, and providing explicit constructions and bounds for such digraphs.

Contribution

It introduces the study of link-irregular digraphs, proves their existence conditions, structural properties, and constructs explicit examples with various properties.

Findings

Link-irregular digraphs exist for n ≥ 5.

Underlying graphs of link-irregular digraphs contain 3-cycles.

Almost all link-irregular digraphs are nonplanar.

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.