Elsevier

Discrete Mathematics

Volume 309, Issue 4, 6 March 2009, Pages 997-1000
Discrete Mathematics

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

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Abstract

It is proved that for all positive integers d,k,s,t with tk+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 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 [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 Kn. 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 (312a+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 result of the present note shows that every graph with edge density at least l(a)+(312a+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.

Keywords

Graph minors
Edge density
Connected subgraphs

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