Spanning trees with at most k leaves in 2-connected K1,r-free graphs

A vertex with degree one and a vertex with degree at least three are called a leaf and a branch vertex in a tree, respectively. In this paper, we obtain that every 2-connected K-1,K-r-free graph G contains a spanning tree with at most k leaves if alpha(G) <= k + inverted right perpendiculark+1/r-3inverted left perpendicular - left perpendicular1/|r-k-3|+1 right perpendicular, where k >= 2 and r >= 4 . The upper bound is best possible. Furthermore, we prove that if a connected K-1,K- 4-free graph G satisfies that alpha (G ) <= 2k + 5 , then G contains either a spanning tree with at most k branch vertices or a block B with alpha (B) <= 2 . A related conjecture for 2-connected claw-free graphs is also posed. (c) 2023 Elsevier Inc. All rights reserved.