Advanced Search

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 $\left(\frac{N}{2}\right)-o\left(N^2\right)$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1-o\left(1\right)}$. 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 $N^{2-\delta }$ induced matchings, where $\delta >0,076$. This disproves (in a strong form) a conjecture of Meshulam, substantially...