Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs

TL;DR

This paper explores the relationships between bandwidth, treewidth, separators, and expansion in bounded degree graphs, showing that sublinear bounds on one imply sublinear bounds on all, with implications for graph universality.

Contribution

It establishes new relations between key graph parameters and provides a criterion for universality in classes like bounded-degree planar graphs.

Findings

Graphs of fixed genus have sublinear bandwidth.

Sublinear parameters imply similar bounds for all related parameters.

Large minimum degree graphs contain all bounded-degree planar graphs.

Abstract

We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these parameters is sublinear in the number of vertices of G then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each gamma>0 every n-vertex graph with minimum degree ((3/4)+gamma)n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large.