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A full Halin grid theorem

Agelos Georgakopoulos, Matthias Hamann

Abstract

Halin's well-known grid theorem states that a graph $G$ with a thick end must contain a subdivision of the hexagonal half-grid. We obtain the following strengthening when $G$ is vertex-transitive and locally finite. Either $G$ is quasi-isometric to a tree (and therefore has no thick end), or it contains a subdivision of the full hexagonal grid.

A full Halin grid theorem

Abstract

Halin's well-known grid theorem states that a graph with a thick end must contain a subdivision of the hexagonal half-grid. We obtain the following strengthening when is vertex-transitive and locally finite. Either is quasi-isometric to a tree (and therefore has no thick end), or it contains a subdivision of the full hexagonal grid.
Paper Structure (11 sections, 13 theorems, 5 equations)

This paper contains 11 sections, 13 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs
  4. Embeddings
  5. Proof of the main result
  6. The planar 1-ended case
  7. The planar multi-ended case
  8. The non-planar case
  9. Diverging subdivisions of $\mathbb H$
  10. Well-quasi-ordering Cayley graph s
  11. Finer minor notions

Key Result

Theorem 1.1

Let $G\ $ be a locally finite, quasi-tran-si-tive graph that is not quasi-iso-metric to a tree. Then $G\ $ contains a subdivision of $\mathbb{H}$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 3.1: GeoPapMin
  • Lemma 3.2
  • proof
  • Theorem 3.3: CanonicalTTD
  • Lemma 3.4: EGL-QuasiTransitiveGraphsAvoidingMinor
  • Theorem 3.5
  • proof
  • Theorem 3.6: EGL-QuasiTransitiveGraphsAvoidingMinor
  • Theorem 3.7
  • ...and 11 more