Nearly complete graphs decomposable into large induced matchings and their applications

Authors:
Noga Alon;Ankur Moitra;Benny Sudakov
Affiliations:
Tel Aviv University and Institute for Advanced Study, Tel Aviv, Israel;Institute for Advanced Study, Princeton, USA;University of California, Los Angeles, Los Angeles, USA
Venue:
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Year:
2012

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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 (N2)-o(N2) edges, which can be decomposed into pairwise disjoint induced matchings, each of size N1-o(1). The 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-δ induced matchings, where δ0.076. 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 Hastad 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.