TY - GEN
T1 - Nearly complete graphs decomposable into large induced matchings and their applications
AU - Alon, Noga
AU - Moitra, Ankur
AU - Sudakov, Benny
PY - 2012
Y1 - 2012
N2 - 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 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.
AB - 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 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.
KW - additive combinatorics
KW - induced matchings
UR - http://www.scopus.com/inward/record.url?scp=84862606987&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84862606987&partnerID=8YFLogxK
U2 - 10.1145/2213977.2214074
DO - 10.1145/2213977.2214074
M3 - Conference contribution
AN - SCOPUS:84862606987
SN - 9781450312455
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 1079
EP - 1089
BT - STOC '12 - Proceedings of the 2012 ACM Symposium on Theory of Computing
T2 - 44th Annual ACM Symposium on Theory of Computing, STOC '12
Y2 - 19 May 2012 through 22 May 2012
ER -