On Zarankiewicz Problem and Depth-Two Superconcentrators

TL;DR

This paper establishes precise conditions on bipartite graph unions to avoid large independent sets, unifying classical theorems and applying these insights to derive tight bounds for depth-two superconcentrators.

Contribution

It generalizes two fundamental theorems in bipartite graph theory and applies these results to improve bounds on depth-two superconcentrators.

Findings

Unified classical theorems on bipartite graphs and independent sets

Derived tight necessary and sufficient conditions for bipartite graph unions

Applied results to establish tight lower bounds for depth-two superconcentrators

Abstract

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical results in graph theory: the theorem of K\H{o}v\'{a}ri, S\'{o}s and Tur\'{a}n on the minimum number of edges in a bipartite graph that has no large independent set, and the theorem of Hansel (also Katona and Szemer\'{e}di, Krichevskii) on the sum of the sizes of bipartite graphs that can be used to construct a graph (non-necessarily bipartite) that has no large independent set. As an application of our results, we show how they unify the underlying combinatorial principles developed in the proof of tight lower bounds for depth-two superconcentrators.