Asymptotic dimension of minor-closed families and beyond
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 compactnes...
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ftdatacite:10.48550/arxiv.2007.08771 2023-05-15T16:31:26+02:00 Asymptotic dimension of minor-closed families and beyond Liu, Chun-Hung 2020 https://dx.doi.org/10.48550/arxiv.2007.08771 https://arxiv.org/abs/2007.08771 unknown arXiv https://dx.doi.org/10.1137/1.9781611976465.119 arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Combinatorics math.CO Discrete Mathematics cs.DM FOS Mathematics FOS Computer and information sciences article-journal Article ScholarlyArticle Text 2020 ftdatacite https://doi.org/10.48550/arxiv.2007.08771 https://doi.org/10.1137/1.9781611976465.119 2022-03-10T15:43:44Z 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. Article in Journal/Newspaper Groenland DataCite Metadata Store (German National Library of Science and Technology) |
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Combinatorics math.CO Discrete Mathematics cs.DM FOS Mathematics FOS Computer and information sciences |
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Combinatorics math.CO Discrete Mathematics cs.DM FOS Mathematics FOS Computer and information sciences Liu, Chun-Hung Asymptotic dimension of minor-closed families and beyond |
topic_facet |
Combinatorics math.CO Discrete Mathematics cs.DM FOS Mathematics FOS Computer and information sciences |
description |
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. |
format |
Article in Journal/Newspaper |
author |
Liu, Chun-Hung |
author_facet |
Liu, Chun-Hung |
author_sort |
Liu, Chun-Hung |
title |
Asymptotic dimension of minor-closed families and beyond |
title_short |
Asymptotic dimension of minor-closed families and beyond |
title_full |
Asymptotic dimension of minor-closed families and beyond |
title_fullStr |
Asymptotic dimension of minor-closed families and beyond |
title_full_unstemmed |
Asymptotic dimension of minor-closed families and beyond |
title_sort |
asymptotic dimension of minor-closed families and beyond |
publisher |
arXiv |
publishDate |
2020 |
url |
https://dx.doi.org/10.48550/arxiv.2007.08771 https://arxiv.org/abs/2007.08771 |
genre |
Groenland |
genre_facet |
Groenland |
op_relation |
https://dx.doi.org/10.1137/1.9781611976465.119 |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.2007.08771 https://doi.org/10.1137/1.9781611976465.119 |
_version_ |
1766021157378588672 |