A necessary and sufficient condition for the existence of a spanning tree with specified vertices having large degrees

Let G be a connected simple graph, let X⊆V (G) and let f be a mapping from X to the set of integers. When X is an independent set, Frank and Gyárfás, and independently, Kaneko and Yoshimoto gave a necessary and sufficient condition for the existence of spanning tree T in G such that dT(x) for all x ∈ X, where dT(x) is the degree of x and T. In this paper, we extend this result to the case where the subgraph induced by X has no induced path of order four, and prove that there exists a spanning tree T in G such that dT (x) ≥ f(x) for all x ∈ X if and only if for any nonempty subset S ⊆ X, |NG(S) − S| − f(S) + 2|S| − ωG(S) ≥, where ωG(S) is the number of components of the subgraph induced by S.