Let k ≥ 2 be an integer. We show that if G is a (k + 1)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G| + 1, then for each subset S of V (G) with |S| = k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore's theorem which guarantees the existence of a Hamilton path connecting any… (More)
For a graph H and an integer k ≥ 2, let σ k (H) denote the minimum degree sum of k independent vertices of H. We prove that if a connected claw-free graph G satisfies σ k+1 (G) ≥ |G| − k, then G has a spanning tree with at most k leaves. We also show that the bound |G| − k is sharp and discuss the maximum degree of the required spanning trees.