Rooted minor problems in highly connected graphs

Abstract 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 means that for any distinct k vertices v 1 ,..., v k , there are connected subgraphs H 1 ,..., H a , K 1 ,..., K k such that each of K i contains v i and is adjacent to all H 1 ,..., 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 K a , k -minor.