Nearly complete graphs decomposable into large induced matchings and their applications

Noga Alon, Ankur Moitra, Benny Sudakov
2012 Proceedings of the 44th symposium on Theory of Computing - STOC '12  
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 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−δ induced matchings, where δ > 0.076. This disproves (in a strong
more » ... 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.
doi:10.1145/2213977.2214074 dblp:conf/stoc/AlonMS12 fatcat:jddake3xkfh2xmnidtqmcuz5ta