A closure result on spanning $k$-trees of graphs with given minimum degree

Abstract

Let $k\geq2$ be an integer. A $k$-tree is a tree with maximum degree at most $k$. In this paper, we give a closure result on spanning $k$-trees of graphs with given minimum degree. Let $\delta\geq1$ be an integer, and $G$ be a connected graph of order $n$ with minimum degree $\delta$. Let $u$ and $v$ be two nonadjacent vertices of $G$ satisfying $d_{G}(u)+d_{G}(v)\geq n-1-(k-2)\delta$. Then $G$ has a spanning $k$-tree if and only if $G+uv$ has a spanning $k$-tree.