Degree Conditions and Degree Bounded Trees

In this paper, we give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and $A?V(G)$. We denote by ${\sigma }_k(A)$ the minimum value of the degree sum in G of any k pairwise nonadjacent vertices of A, and by $w(G?A)$ the number of components of the subgraph $G?A$ of G induced by $V(G)?A$. Our main results are the following: (i) If ${\sigma }_k(A)?|G|?1$, then G contains a tree T with maximum degree ?k and $A?V(T)$. (ii) If ${\sigma }_{k?w(G?A)}(A)?|A|?1$, then G contains a spanning tree T with $d_T(x)?k$ for any $x\in A$. These are generalizations of the result by S. Win S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Seminar Univ. Humburg 43 (1975) 263–267] and degree conditions are sharp.