Halin's grid theorem for digraphs

TL;DR

This paper extends Halin's grid theorem from undirected graphs to directed graphs, establishing the existence of grid structures within certain infinite families of rays in digraphs.

Contribution

It generalizes Halin's theorem to digraphs, showing that specific infinite families of rays contain grid-like substructures.

Findings

Existence of grids in digraphs with infinite families of rays

Results for in-rays and necklaces similar to out-rays

Extension of Halin's theorem to directed graphs

Abstract

Halin showed that every thick end of every graph contains an infinite grid. We extend Halin's theorem to digraphs. More precisely, we show that for every infinite family $\mathcal{R}$ of disjoint equivalent out-rays there is a grid whose vertical rays are contained in $\mathcal{R}$. Furthermore, we obtain similar results for in-rays and necklaces.