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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
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.
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.
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