Decomposing a Graph Into Expanding Subgraphs

TL;DR

This paper demonstrates that certain bounds in decomposing graphs into expanding subgraphs are essentially optimal, using new graph constructions based on random hypercube subgraphs and metric embedding techniques.

Contribution

The paper introduces a new family of graphs derived from random hypercube subgraphs, establishing the optimality of bounds in graph decomposition into expanders.

Findings

Bounds in graph decomposition are tight and essentially optimal.

New graph constructions based on hypercube subgraphs support these bounds.

Analysis employs simple metric embedding arguments.

Abstract

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. These results are obtained as corollaries of a new family of graphs, which we construct by picking random subgraphs of the hypercube, and analyze using (simple) arguments from the theory of metric embedding.