Many disjoint dense subgraphs versus large [formula omitted]-connected subgraphs in large graphs with given edge density

Abstract

It is proved that for all positive integers d , k , s , t with t ≥ k + 1 there is a positive integer M = M ( d , k , s , t ) such that every graph with edge density at least d + k and at least M vertices contains a k -connected subgraph on at least t vertices, or s pairwise disjoint subgraphs with edge density at least d . By a classical result of Mader [W. Mader, Existenz n -fach zusammenhängender Teilgraphen in Graphen genügend großer Kantendichte, Abh. Math. Sem Univ. Hamburg, 37 (1972) 86–97] this implies that every graph with edge density at least 3 k 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 [W. Mader, Homomorphiesätze für Graphen, Math. Ann. 178 (1968) 154–168] 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 K n . Recently, it was proved in [T. Böhme, K. Kawarabayashi, J. Maharry, B. Mohar, Linear connectivity forces dense minors, J. Combin. Theory Ser. B (submitted for publication)] that every ( 31 2 a + 1 ) -connected graph with sufficiently many vertices either has a topological minor isomorphic to K a , p q , or it has a minor isomorphic to the disjoint union of p copies of K a , q . Combining these results with the 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 K a , p a , or a minor isomorphic to the disjoint union of p copies of K a . This implies an affirmative answer to a question of Fon-der-Flaass.