On a k-tree containing specified leaves in a graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k ≥ 2, 0 ≤ s ≤ k, and n ≤ s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k-1)s-1, or (2) G is n-connected and the independence number of G is at most (n-s)(k-1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results.