Another equivalent of the graceful tree conjecture

Let T = (V, E) be a tree on \V\ = n vertices. T is graceful if there exists a bijection f : V --> (0, 1,..., n - 1) such that {\f(u) - f(V)\ uv is an element of E} = (1,2,...,n - 1). If, moreover, T contains a perfect matching M and f can be chosen in such a way that f(u) + f(v) = n - 1 for every edge uv is an element of M (implying that {\f(u) - f(v)\ I u u is an element of M} = {1, 3,..., n - }), then T is called strongly graceful. We show that the well-known conjecture that all trees are graceful is equivalent to the conjecture that all trees containing a perfect matching are strongly graceful. We also give some applications of this result.