Chun-Hung Liu

Asymptotic dimension of minor-closed families, and beyond.

We consider the asymptotic dimension of the metric spaces given by the
shortest path metric of (edge-)weighted graphs. We prove that the
asymptotic dimension of any minor-closed family of weighted graphs, any
class of weighted 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 generalize and improve some results in the
literature, including results for Riemannian surfaces and Cayley graphs
of groups with a forbidden minor.