Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density

It is proved that every graph with edge density at least d + k and sufficiently many vertices contains a k-connected subgraph on at least t (t ≥ k+1) vertices, or s pairwise disjoint subgraphs with edge density at least d. By a classical result of Mader [11] this implies that every graph with edge density at least 3k and sufficiently many vertices contains a k-connected subgraph with at least r vertices, or r pairwise disjoint k-connected subgraphs. Another classical result of Mader [10] states that for every n there is an l(n) such that every graph with edge density at least l(n) contains a minor isomorphic to Kn. Recently, it was proved in [1] that every ( 31 2 a+1)-connected graph with sufficiently many vertices either has a topological minor isomorphic to Ka,pq, or it has a minor isomorphic to the disjoint union of p copies of Ka,q. Combining these results with the main result of the present note shows that every graph with edge density at least l(a) + ( 31 2 a+ 1) and sufficiently many vertices has a topological minor isomorphic to Ka,pa, or a minor isomorphic to the disjoint union of p copies of Ka. This implies an affirmative answer to a question of Fon-der-Flaass. ∗Research of this author is supported in part by the NSF grant DMS-0400498 and grant 03-01-00796 of the Russian Foundation for Fundamental Research