Asymptotic dimension of minor-closed families and beyond

Abstract

The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics are the distance functions in graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite graphs.In this paper we prove that the asymptotic dimension of any proper minor-closed family, any class of graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1, and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and improve a number of results in the literature. Our proofs can be transformed into linear or quadratic time algorithms for finding coverings witnessing the asymptotic dimension which is equivalent to finding weak diameter colorings for graphs.The key ingredient of our proof is a unified machinery about the asymptotic dimension of classes of graphs that have tree-decompositions of bounded adhesion over hereditary classes with known asymptotic dimension, which might be of independent interest.

Surfaces have (asymptotic) dimension 2

“…Note added. After we made our manuscript public, Liu [24] gave a positive answer to Question 1, in turn implying positive answers to Questions 3, 4, 5, 6, 7, and 9. are (R 2 + 2r 2 , s 2 + 2s 2 )-bounded. Since (r 1 , s 1 ) = (r, r), (r 1 , s 1 )-components of D j are the same as r-components of D j , so it follows that r-components of D j are (R 2 + 2r 2 )-bounded.…”

Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces

“…Recall that this paper is a combination of parts of [10,29], where a number of results in [10] are strengthened in this paper.…”

“…Note that positive answers of both Questions 3 and 4 would give a characterization of monotone graph classes with bounded asymptotic dimension in terms of polynomial expansion. Question 4 is inspired by a question of David Wood in private communication after preliminary forms of this paper [10,29] appeared in arXiv. He asked whether every monotone class of graphs of bounded asymptotic dimension has bounded expansion.…”

“…This paper is a combination of parts of two manuscripts[10,29] in the arXiv repository, where some results in[10] are strengthened in this paper. Theorem 4.3 of this paper is included in the first version of[10] and is an immediate corollary of a combination of results in[11,20], as shown in[10].…”

Graph Colouring and Decomposition

“…A key property is that layered treewidth is bounded on planar graphs but treewidth is not. In particular, planar graphs have layered treewidth at most 3 [27], but the n×n grid graph has treewidth n. Layered treewidth has been used in upper bounds on several graph parameters including queue-number [25,27], stack number [25], boxicity [69], clustered chromatic number [50], generalised colouring numbers [73], asymptotic dimension [8,48], as well for results in intersection graph theory [71]. Row treewidth 7 is a reőnement of layered treewidth, implicitly introduced by Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [26], who proved that planar graphs have row treewidth at most 8.…”