On a Spanning $k$-Tree in which Specified Vertices Have Degree Less Than $k$

A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a degree sum condition for a graph to have a spanning $k$-tree in which specified vertices have degree less than $k$. We denote by $\sigma_k(G)$ the minimum value of the degree sum of $k$ independent vertices in a graph $G$. Let $k ≥ 3$ and s $≥ 0$ be integers, and suppose $G$ is a connected graph and $\sigma_k(G) ≥ |V (G)|+s−1$. Then for any $s$ specified vertices, $G$ contains a spanning $k$-tree in which every specified vertex has degree less than $k$. The degree condition is sharp.