A Note on Leaf-constrained Spanning Trees in a Graph

An independent set S of a connected graph G is called a frame if G − S is connected. If |S| = k, then S is called a k-frame. We prove the following theorem. Let k ≥ 2 be an integer, G be a connected graph with V (G) = {v1, v2, . . . , vn}, and degG(u) denote the degree of a vertex u. Suppose that for every 3-frame S = {va, vb, vc} such that 1 ≤ a < b < c ≤ n, degG(va) ≤ a, degG(vb) ≤ b − 1 and degG(vc) ≤ c − 2, it holds that degG(va) + degG(vb) + degG(vc) −|NG(va)∩NG(vb)∩NG(vc)| ≥ |G| − k +1. Then G has a spanning tree with at most k-leaves. Moreover, the condition is sharp. This theorem is a generalization of the results of E. Flandrin, H.A. Jung and H. Li (Discrete Math. 90 (1991), 41–52) and of A. Kyaw (Australasian Journal of Combinatorics. 37 (2007), 3–10) for traceability.