Trees through specified vertices

We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n − 2 if G is a connected graph with A ⊂ V ( G ) such that d G ( v ) ≥ k for all v ∈ A , then there exists a subtree T of G such that V ( T ) ⊃ A and d T ( v ) ≤ ⌈ n − 1 k ⌉ for all v ∈ A .