Rooted minor problems in highly connected graphs

The purpose of this note is to give a connectivity condition for a graph to have a rooted complete bipartite minor. Here, a rooted complete bipartite graph minor K-a,K-k means that for any distinct k vertices v(1),...v(k), there are connected subgraphs H-1,..., H-a, K-l,..., K-k such that each of K-i contains v(i) and is adjacent to all H-l,..., H-a. Roughly, our results say that if G is large enough, then the linear connectivity on the function of k guarantees the existence of a rooted K-a,(k)-minor (for any a), and in general, the connectivity condition on the existence of a rooted K-a,(k) -minor is "almost" the same as the average degree which forces the existence of a Ka,k-minor. (C) 2004 Elsevier B.V All rights reserved.