A spanning tree with at most k leaves in a K1,5-free graph

Abstract

A tree is called a k-ended tree if it has at most k leaves. Let $k\ge 2$ and $p\ge 3$ be integers, let G be a connected $K_{1,p}$-free graph, and let ${\sigma }_{k+1}\left(G\right)$ be the minimum degree sum of pair-wisely non-adjacent $k+1$ vertices of G. For $p=3,4$ or for $p=5$ and $k=4,5,6$, the lower bounds of ${\sigma }_{k+1}\left(G\right)$ which assure the existence of spanning k-ended trees are known. In this paper, we extend these results to the case $p=5$ and any $k\ge 2$, which states that for a connected $K_{1,p}$-free graph, if $k\ge 4$ and ${\sigma }_{k+1}\left(G\right)\ge \left|G\right|-k/3$, or if $k=3$ and ${\sigma }_{k+1}\left(G\right)\ge \left|G\right|$, or if $k=2$, $\left|G\right|\ge 7$ and ${\sigma }_{k+1}\left(G\right)\ge \left|G\right|$, then G has a spanning k-ended tree. These lower bounds of the assumptions are best possible.