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A Full Halin Grid Theorem.

Agelos Georgakopoulos¹, Matthias Hamann²

¹Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK.

Discrete & Computational Geometry

|June 2, 2026

Summary

This summary is machine-generated.

Vertex-transitive graphs with a thick end either resemble trees or contain a hexagonal grid subdivision. This research strengthens Halin's grid theorem for specific graph types.

Keywords:

Halin’s grid theorem Quasi-isometry Thick end Vertex-transitive Well-quasi-ordering

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Area of Science:

Graph theory
Geometric group theory

Background:

Halin's grid theorem: Graphs with a thick end contain a hexagonal half-grid subdivision.
Vertex-transitive graphs possess high symmetry.
Locally finite graphs have finite neighborhoods for each vertex.

Purpose of the Study:

To strengthen Halin's grid theorem for vertex-transitive, locally finite graphs.
To investigate the structural properties of such graphs concerning grid subdivisions.

Main Methods:

Utilizing quasi-isometry to classify graph structures.
Applying concepts from geometric group theory to analyze graph ends.
Proving structural dichotomy based on graph properties.

Main Results:

A vertex-transitive, locally finite graph either is quasi-isometric to a tree or contains a subdivision of the full hexagonal grid.
Graphs quasi-isometric to trees lack a thick end, satisfying the theorem's condition vacuously.

Conclusions:

The study provides a refined understanding of graph structure, particularly for vertex-transitive graphs.
The findings offer a stronger alternative to Halin's theorem under specific graph conditions.