Tight conditions for spanning trees with leaf degree at most k in graphs

Abstract

Let $k$ be a positive integer, $G$ be a connected graph of order $n$, and $T$ be a tree. For $v\in V\left(T\right)$, the leaf degree of $v$ is defined as $d_{leaf}\left(v\right)=\left|\{u\in N_T\left(v\right)\mid d_T\left(u\right)=1\}\right|$. The leaf degree of $T$ is defined as $d_{leaf}\left(T\right)=max\{d_{leaf}\left(v\right)\mid v\in V\left(T\right)\}$. In this paper, motivated by the structure condition of Kaneko (2001), we obtain some tight conditions in $G$ with respect to the size or the spectral radius to ensure that $G$ has a spanning tree $T$ with $d_{leaf}\left(T\right)\le k$, which improves the result of [2].