Infinite grids in digraphs

TL;DR

This paper extends a classical graph theory result to directed graphs, showing that certain infinite digraphs contain grid-like subdivisions aligned with a specific end, revealing structural properties of infinite digraphs.

Contribution

It proves a new theorem for digraphs, generalizing Halin's result from undirected graphs to directed graphs with infinite rays.

Findings

Infinite digraphs with an end containing infinitely many disjoint directed rays contain grid-like subdivisions.

The result generalizes Halin's theorem from undirected to directed graphs.

Provides insight into the structure of infinite digraphs with complex end behavior.

Abstract

Halin proved that every graph with an end $\omega$ containing infinitely many pairwise disjoint rays admits a subdivision of the infinite quarter-grid as a subgraph where all rays from that subgraph belong to $\omega$. We will prove a corresponding statement for digraphs, that is, we will prove that every digraph that has an end with infinitely many pairwise disjoint directed rays contains a subdivision of a grid-like digraph all of whose directed rays belong to that end.