Nearly Complete Graphs Decomposable into Large Induced Matchings and their Applications

TL;DR

This paper presents new dense graph constructions decomposable into large induced matchings, with applications in communication complexity, graph testing, and Steiner tree problems, disproving conjectures and improving previous bounds.

Contribution

Introduces two novel dense graph constructions decomposable into large induced matchings, impacting multiple areas including graph theory, communication complexity, and combinatorial optimization.

Findings

Constructed graphs with nearly complete edges decomposable into large induced matchings.

Disproved Meshulam's conjecture with dense graphs covered by few induced matchings.

Improved bounds in communication over shared channels and extended analysis of graph testing methods.

Abstract

We describe two constructions of (very) dense graphs which are edge disjoint unions of large {\em induced} matchings. The first construction exhibits graphs on $N$ vertices with ${N \choose 2}-o(N^2)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o(1)}$. The second construction provides a covering of all edges of the complete graph $K_N$ by two graphs, each being the edge disjoint union of at most $N^{2-\delta}$ induced matchings, where $\delta > 0.058$. This disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linial and Meshulam on communicating over a shared channel, and (slightly) extends the analysis of H{\aa}stad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity. Additionally, our constructions settle a combinatorial question of Vempala regarding a candidate rounding scheme for the directed Steiner tree problem.