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Counterexample to the conjectured coarse grid theorem

Sandra Albrechtsen, James Davies

Abstract

We show that for every $M,A,n \in \mathbb{N}$ there exists a graph $G$ that does not contain the $(154\times 154)$-grid as a $3$-fat minor and is not $(M,A)$-quasi-isometric to a graph with no $K_n$ minor. This refutes the conjectured coarse grid theorem by Georgakopoulos and Papasoglu and the weak fat minor conjecture of Davies, Hickingbotham, Illingworth, and McCarty. Our construction is a slight modification of the recent counterexample to the weak coarse Menger conjecture from Nguyen, Scott and Seymour. We further modify the construction to show that there are planar graphs that do not have the coarse Erdős-Pósa property.

Counterexample to the conjectured coarse grid theorem

Abstract

We show that for every there exists a graph that does not contain the -grid as a -fat minor and is not -quasi-isometric to a graph with no minor. This refutes the conjectured coarse grid theorem by Georgakopoulos and Papasoglu and the weak fat minor conjecture of Davies, Hickingbotham, Illingworth, and McCarty. Our construction is a slight modification of the recent counterexample to the weak coarse Menger conjecture from Nguyen, Scott and Seymour. We further modify the construction to show that there are planar graphs that do not have the coarse Erdős-Pósa property.

Paper Structure

This paper contains 11 sections, 13 theorems, 7 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distances
  4. Fat minors
  5. Quasi-isometries
  6. Separations and tree-decompositions
  7. Construction of the counterexample
  8. No large fat grid minors
  9. No quasi-isometry to a Kn-minor-free graph
  10. Counterexample to the coarse Erdős-Pósa property for planar graphs
  11. Concluding remarks

Key Result

Theorem 1

For every $M, A, n \in \mathbb{N}$ with $M \geqslant 1$, there exists a graph $G$ with the following properties:

Figures (8)

  • Figure 1: Depicted is the graph $G_{4,d,2}$. The dotted lines represent paths of length $d+1$.
  • Figure 2: Depicted is the graph $G_{3,d,3}$, except that $r$ should be identified with the roots of the $H_i^2$'s. The dotted lines represent paths of length $d+1$. The green subgraph is $\Delta(x)$.
  • Figure 3: A sketch of the situation in the proof of \ref{['cor:NoTwoPathsBetweenSameAdhesion']}. Every path in $G$ between $\{s_{k_1},s_{k_2}\}$ and $\{s_{\ell_1},s_{\ell_2}\}$ has to intersect $\mathop{\mathrm{\mathbf{TT}}}\nolimits(H') \cup \{\mathop{\mathrm{R}}\nolimits(G)\}$.
  • Figure 4: Indicated in red is the bag $V_x$ from \ref{['constr:TreeDecomp1']} for the case $x=r$ and $h, m = 3$.
  • Figure 5: Depicted is the graph $X$ except that each of the three grids $M_1, M_2, M_3$ should be a $(154\times 154)$-grid (instead of a $(7 \times 7)$-grid).
  • ...and 3 more figures

Theorems & Definitions (40)

  • Conjecture 1.1: Coarse Grid Theorem
  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • ...and 30 more