Bandwidth, treewidth, separators, expansion, and universality

Julia Böttcher, Klaas P. Pruessmann, Anusch Taraz, Andreas Würfl
2008 Electronic Notes in Discrete Mathematics  
We prove that planar graphs with bounded maximum degree have sublinear bandwidth. As a consequence for each γ > 0 every n-vertex graph with minimum degree ( 3 4 + γ)n contains a copy of every bounded-degree planar graph on n vertices. The proof relies on the fact that planar graphs have small separators. Indeed, we show more generally that for any class of bounded-degree graphs the concepts of sublinear bandwidth, sublinear treewidth, the absence of big expanders as subgraphs, and the existence of small separators are equivalent. 1
doi:10.1016/j.endm.2008.06.018 fatcat:dk73isgg5vgsdoy5lkvzgdtmli