Nearly complete graphs decomposable into large induced matchings and their applications

Abstract

We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on $N$ vertices with $\left(\genfrac{}{}{0ex}{}{N}{2}\right)-o(N^2)$ 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 $\delta >0.076$. 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 Håstad 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.