Minimum degree and pan--linked graphs

For a k -linked graph G and a vector S → of 2 k distinct vertices of G , an S → -linkage is a set of k vertex-disjoint paths joining particular vertices of S → . Let T denote the minimum order of an S → -linkage in G . A graph G is said to be pan- k -linked if it is k -linked and for all vectors S → of 2 k distinct vertices of G , there exists an S → -linkage of order t for all t such that T ≤ t ≤ | V ( G ) | . We first show that if k ≥ 1 and G is a graph on n vertices with n ≥ 5 k − 1 and δ ( G ) ≥ n + k 2 , then any nonspanning path system consisting of k paths, one of which has order four or greater, is extendable by one vertex. We then use this to show that for k ≥ 2 and n ≥ 5 k − 1 , a graph on n vertices satisfying δ ( G ) ≥ n + 2 k − 1 2 is pan- k -linked. In both cases, the minimum degree result is shown to be best possible.