Spanning k-Ended Tree in 2-Connected Graph.

Win proved a very famous conclusion that states the graph G with connectivity ? (G), independence number a (G) and a (G) = ?(G) + k - 1 (k = 2) contains a spanning k-ended tree. This means that there exists a spanning tree with at most k leaves. In this paper, we strengthen the Win theorem to the following: Let G be a simple 2-connected graph such that |V(G) | = 2? (G) + k, a (G) = ?(G) + k (k = 2) and the number of maximum independent sets of cardinality ? + k is at most n - 2? - k + 1. Then, either G contains a spanning k-ended tree or a subgraph of K-? V ((k + ? - 1)K1U Kn-2?-k+1).