Spanning Trees of a 𝐾1,5-Free Graph with at Most Seven Leaves and Branch Vertices in Total

Let 𝐺 be a connected 𝐾 1,5 -free graph with 𝑛 vertices. In this paper, we study some optimal sufficient conditions for a connected 𝐾 1,5 -free graph to have a spanning tree with few leaves and branch vertices in total. In particular, we first prove that if 𝜎 5 (𝐺) ≥ 𝑛 − 2, then 𝐺 contains a spanning tree with at most seven leaves and branch vertices. After that, we show all graphs 𝐺 which have no spanning tree with at most seven leaves and branch vertices and 𝜎 5 (𝐺) = 𝑛 − 3.