Summary: Nearly Complete Graphs Decomposable into Large Induced
Matchings and their Applications
November 5, 2011
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 N
which can be decomposed into pairwise disjoint induced matchings, each of size N1-o(1)
second construction provides a covering of all edges of the complete graph KN by two graphs,
each being the edge disjoint union of at most N2-
induced matchings, where > 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°astad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity.
Additionally, our constructions settle a combinatorial question of Vempala regarding a candidate