Spanning trees with at most [formula omitted] leaves in 2-connected [formula omitted]-free graphs

Abstract

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 K1,r-free graph G contains a spanning tree with at most k leaves if α(G)≤k+⌈k+1r−3⌉−⌊1|r−k−3|+1⌋, where k≥2 and r≥4. The upper bound is best possible. Furthermore, we prove that if a connected K1,4-free graph G satisfies that α(G)≤2k+5, then G contains either a spanning tree with at most k branch vertices or a block B with α(B)≤2. A related conjecture for 2-connected claw-free graphs is also posed.