Spanning trees with leaf distance at least four

For a graph G, we denote by i(G) the number of isolated vertices of G. We prove that for a connected graph G of order at least five, if i(G − S) < |S| for all ∅ = S ⊆ V (G), then G has a spanning tree T such that the distance in T between any two leaves of T is at least four. This result was conjectured by Kaneko in ”Spanning trees with constrains on the leaf degree”, Discrete Applied Math., 115 (2001) 73–76. Moreover, the condition in the result is sharp in a sense that the condition i(G− S) < |S| cannot be replaced by i(G− S) ≤ |S|.