Degree conditions and degree bounded trees

We give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and A a vertex subset of G . We denote by σ k ( A ) the minimum value of the degree sum in G of any k independent vertices in A and by w ( G − A ) the number of components in the induced subgraph G − A . Our main results are the following: (i) If σ k ( A ) ≥ | V ( G ) | − 1 , then G contains a tree T with maximum degree at most k and A ⊆ V ( T ) . (ii) If σ k − w ( G − A ) ( A ) ≥ | A | − 1 , then G contains a spanning tree T such that d T ( x ) ≤ k for every x ∈ A . These are generalizations of the result by Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Sem. Univ. Hamburg 43 (1975) 263–267] and the degree conditions are sharp.