Finding bipartite subgraphs efficiently

TL;DR

This paper presents polynomial algorithms for efficiently finding and decomposing large bipartite subgraphs in graphs, improving upon previous non-constructive existence proofs.

Contribution

It introduces the first polynomial algorithms for constructing bipartite subgraphs and decompositions, previously known only through non-constructive proofs.

Findings

Polynomial algorithms for finding complete bipartite subgraphs.

Efficient decomposition of graphs into bipartite components.

Improved bounds on bipartite subgraph sizes.

Abstract

Polynomial algorithms are given for the following two problems: given a graph with $n$ vertices and $m$ edges, where $m \ge 3 n^{3/2}$, find a complete balanced bipartite subgraph with parts about $\ln n/(\ln (n^2/m))$, given a graph with $n$ vertices, find a decomposition of its edges into complete balanced bipartite graphs having altogether $O(n^2 / \ln n)$ vertices. Previous proofs of the existence of such objects, due to K\H{o}v\'ari-S\'os-Tur\'an, Chung-Erd\H{o}s-Spencer, Bublitz and Tuza were non-constructive.