Spanning trees with at most 4 leaves in $K_{1,5}-$free graphs

Abstract

In 2009, Kyaw proved that every $n$-vertex connected $K_{1,4}$-free graph $G$ with ${\sigma }_4(G)\ge n-1$ contains a spanning tree with at most $3$ leaves. In this paper, we prove an analogue of Kyaw's result for connected $K_{1,5}$-free graphs. We show that every $n$-vertex connected $K_{1,5}$-free graph $G$ with ${\sigma }_5(G)\ge n-1$ contains a spanning tree with at most $4$ leaves. Moreover, the degree sum condition `${\sigma }_5(G)\ge n-1$' is best possible.