High powers of grids in random geometric graphs and thresholds for wide spanning structures

Abstract

Using results of Leighton and Shor and of Shor and Yukich, we present a connection between $d$-dimensional bandwidth and (sharp) thresholds in $d$-dimensional random geometric graphs: namely, that the sharp threshold for a given graph property in random geometric graphs is determined by its $d$-dimensional bandwidth, and vice versa, whenever the $d$-dimensional bandwidth of this property grows sufficiently fast (at least polylogarithmically) with the order of the graphs. This connection provides a number of corollaries, such as new information about the sharp thresholds for containing different spanning subgraphs in $1$-dimensional random geometric graphs, as well as asymptotically tight bounds for the $d$-dimensional bandwidth of structures such as graphs of bounded treewidth or genus, giving an asymptotic resolution to several problems raised by Chung in 1988.