Ubiquity of oriented rays

TL;DR

This paper characterizes which directed graphs with an underlying undirected structure of a ray are ubiquitous, meaning they appear infinitely often in larger digraphs containing arbitrarily many disjoint copies.

Contribution

It provides a complete characterization of the ubiquity property for digraphs whose underlying undirected graph is a ray.

Findings

Identifies all rays that are ubiquitous in digraphs.

Establishes conditions under which a ray is ubiquitous.

Advances understanding of pattern repetition in directed graphs.

Abstract

Call a digraph $H$ \emph{ubiquitous} if every digraph $D$ that contains $k$ vertex-disjoint copies of $H$ for every $k \in \mathbb{N}$ also contains infinitely many vertex-disjoint copies of $H$. We characterise which digraphs whose underlying undirected graph is a ray are ubiquitous.