Geodesic
Nearly complete graphs decomposable into large induced matchings and their applications
Journal of the European Mathematical Society, Tome 15 (2013) no. 5, pp. 1575-1596 Cet article a éte moissonné depuis la source EMS Press

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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 (2N​)−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 δ>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.
DOI : 10.4171/jems/398
Classification : 05-XX, 68-XX, 00-XX
Keywords: Ruzsa-Szemerédi graphs, induced matchings, testing linearity 
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     title = {Nearly complete graphs decomposable into large induced matchings and their applications},
     journal = {Journal of the European Mathematical Society},
     pages = {1575--1596},
     year = {2013},
     volume = {15},
     number = {5},
     doi = {10.4171/jems/398},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/398/}
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Noga Alon; Ankur Moitra; Benjamin Sudakov. Nearly complete graphs decomposable into large induced matchings and their applications. Journal of the European Mathematical Society, Tome 15 (2013) no. 5, pp. 1575-1596. doi: 10.4171/jems/398

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