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

A tree is called a k-ended tree if it has at most k leaves. Let k >= 2 and p >= 3 be integers, let G be a connected K1,p-free graph, and let sigma k+1(G) be the minimum degree sum of pairwisely 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(G) 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 >= 2, which states that for a connected K1,p-free graph, if k >= 4 and sigma k+1(G) >= | G | - k/3, or if k = 3 and sigma k+1(G) >= | G |, or if k = 2, |G| >= 7 and sigma k+1(G) >= |G|, then G has a spanning k-ended tree. These lower bounds of the assumptions are best possible. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.