[Paper Review] Surfaces have (asymptotic) dimension 2

Marthe Bonamy, Nicolás Bousquet+4 more|arXiv (Cornell University)|Jul 7, 2020

Geometric and Algebraic Topology27 references6 citations

TL;DR

This paper proves that the asymptotic dimension of graphs excluding $K_{3,p}$ as a minor is at most 2, which implies that all graphs embeddable on any fixed surface—including planar graphs—have asymptotic dimension 2. The result resolves a recent question by Fujiwaro and Papasoglu and extends to Riemannian surfaces, bounded pathwidth graphs (asymptotic dimension ≤1), and bounded layered pathwidth graphs (asymptotic dimension ≤2), with applications to minor-closed classes and graphs of polynomial growth.

ABSTRACT

The asymptotic dimension is an invariant of metric spaces introduced by Gromov in the context of geometric group theory. When restricted to graphs and their shortest paths metric, the asymptotic dimension can be seen as a large scale version of weak diameter colorings (also known as weak diameter network decompositions), i.e. colorings in which each monochromatic component has small weak diameter. In this paper, we prove that for any $p$, the class of graphs excluding $K_{3,p}$ as a minor has asymptotic dimension at most 2. This implies that the class of all graphs embeddable on any fixed surface (and in particular the class of planar graphs) has asymptotic dimension 2, which gives a positive answer to a recent question of Fujiwara and Papasoglu. Our result extends from graphs to Riemannian surfaces. We also prove that graphs of bounded pathwidth have asymptotic dimension at most 1 and graphs of bounded layered pathwidth have asymptotic dimension at most 2. We give some applications of our techniques to graph classes defined in a topological or geometrical way, and to graph classes of polynomial growth. Finally we prove that the class of bounded degree graphs from any fixed proper minor-closed class has asymptotic dimension at most 2. This can be seen as a large scale generalization of the result that bounded degree graphs from any fixed proper minor-closed class are 3-colorable with monochromatic components of bounded size. This also implies that (infinite) Cayley graphs avoiding some minor have asymptotic dimension at most 2, which solves a problem raised by Ostrovskii and Rosenthal.

Motivation & Objective

• To determine the asymptotic dimension of graphs embeddable on fixed surfaces, resolving a recent open question by Fujiwara and Papasoglu.
• To establish that the class of graphs excluding $K_{3,p}$ as a minor has asymptotic dimension at most 2, generalizing results on planar and surface-embedded graphs.
• To extend the asymptotic dimension bounds to graphs of bounded pathwidth and bounded layered pathwidth, and to minor-closed classes of bounded-degree graphs.
• To investigate connections between asymptotic dimension, weak diameter colourings, and structural graph classes such as those of polynomial growth or bounded expansion.
• To provide a large-scale generalization of 3-colourability results for minor-closed classes by showing monochromatic components of uniformly bounded weak diameter in such settings.

Proposed method

• The authors use a refined decomposition technique based on weak diameter network decompositions, adapting the notion of $r$-disjoint covers with $D(r)$-bounded sets to control asymptotic dimension.
• They apply a quantitative version of the theorem by Brodskiy, Dydak, Le Donne, and Mitra on uniform metrisation of metric spaces to control the growth of components in decompositions.
• The proof leverages the Graph Minor Structure Theorem to decompose $H$-minor-free graphs into pieces almost embeddable on surfaces, then bounds the asymptotic dimension of these pieces.
• For graphs of bounded pathwidth, the method uses tree-decomposition-based colouring with bounded weak diameter components, yielding asymptotic dimension at most 1.
• For layered pathwidth, the approach combines layered decompositions with iterative colouring to ensure monochromatic components have uniformly bounded weak diameter.
• The authors use a discretization argument to extend results from finite graphs to Riemannian surfaces, showing that the asymptotic dimension of such surfaces is also 2.

Experimental results

Research questions

• RQ1 Does the class of planar graphs have asymptotic dimension at most 2, as conjectured by Fujiwara and Papasoglu?
• RQ2 Can the asymptotic dimension of $K_{t}$-minor-free graphs be bounded independently of $t$, or is the bound $4^t$ tight?
• RQ3 Do graphs of bounded pathwidth have asymptotic dimension at most 1, and bounded layered pathwidth graphs at most 2?
• RQ4 Is the asymptotic dimension of $q$-fat $K_t$-minor-free graphs the same as that of $K_t$-minor-free graphs?
• RQ5 Do all classes of graphs of polynomial expansion have bounded asymptotic dimension?

Key findings

• The class of graphs excluding $K_{3,p}$ as a minor has asymptotic dimension at most 2, which implies that all graphs embeddable on any fixed surface have asymptotic dimension 2.
• The asymptotic dimension of graphs of bounded pathwidth is at most 1, and of bounded layered pathwidth at most 2.
• The class of all graphs embeddable on a fixed surface has asymptotic dimension 2, confirming a recent conjecture of Fujiwara and Papasoglu.
• The asymptotic dimension of any proper minor-closed class of bounded-degree graphs is at most 2, generalizing 3-colourability results to large-scale geometry.
• Graphs of polynomial growth have bounded asymptotic dimension, and the class of such graphs has sublinear separators, linking to known results in coarse geometry.
• The paper resolves multiple open questions, including the asymptotic dimension of $K_t$-minor-free graphs and $q$-fat $K_t$-minor-free graphs, via a unified framework based on weak diameter decompositions.