Spanning trees with leaf distance at least d

The leaf distance of a tree is the maximum d such that the distance between any pair of leaves in the tree is at least d. Kaneko provided sufficient conditions to force the existence of a spanning tree with leaf distance at least d=3 and conjectured that similar conditions suffice for larger d. The case when d=4 was later proved by Kaneko, Kano, and Suzuki. In this paper, we show that when d≥4, a stronger form of this conjecture holds for graphs with independence number at most five. As an immediate corollary, we obtain that when d≥n∕3, this stronger version holds for all n-vertex graphs, consequently proving Kaneko's conjecture for d≥n∕3.