Heavy cycles and spanning trees with few leaves in weighted graphs

Let G be a 2-connected weighted graph and k ≥ 2 an integer. In this note we prove that if the sum of the weighted degrees of every k + 1 pairwise nonadjacent vertices is at least m , then G contains either a cycle of weight at least 2 m / ( k + 1 ) or a spanning tree with no more than k leaves.