Infinite grids in digraphs

Abstract

Halin proved that every graph with an end $ω$ 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 $ω$. 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.

Figures (6)

• Figure 1.1: The hexagonal quarter-grid $H^{\infty}$.
• Figure 1.2: The bidirected quarter-grid.
• Figure 4.1: The cyclically directed quarter-grids.
• Figure 4.2: A subdivision of the bidirected quarter-grid in the ascending (on the left) and the descending (on the right) cyclically directed quarter-grid, where the edges coloured in blue, red and orange correspond to subdivisions of the rays $R_1$, $R_2$, and $R_3$ from the bidirected quarter-grid. The cyan edges highlight the edges of bidirected quarter-grid between the rays.
• Figure 4.3: The digraph from Zuther Z1998 with the red ray being from the thick end.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof : Proof of Theorem \ref{['thm:Kasper']}.
• Lemma 4.1