The  leaf distance    of a tree is the maximum   d     d        such that the distance between any pair of leaves in the tree is at least   d     d       . Kaneko provided sufficient conditions to force the existence of a spanning tree with leaf distance at least   d=3     d  =  3        and conjectured that similar conditions suffice for larger   d     d       . The case when   d=4     d  =  4        was later proved by Kaneko, Kano, and Suzuki. In this paper, we show that when   d≥4     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     d  ≥  n  ∕  3       , this stronger version holds for all   n     n       -vertex graphs, consequently proving Kaneko’s conjecture for   d≥n∕3     d  ≥  n  ∕  3       .
['Catherine Erbes', 'Theodore Molla', 'Sarah Mousley', 'Michael Santana']
Spanning trees with leaf distance at least d