Almost all trees are almost graceful

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each c>0 and for all n>n_0(c). Suppose that (i) the maximum degree of T is bounded by O(n/log n), and (ii) the vertex labels are chosen from the set {1,2,..., (1+c)n}. Then there is an injective labelling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labelling. As a consequence, for any such tree T we can pack (2+2c)n-1 copies of T into the complete graph of order (2+2c)n-1 cyclically. This proves an approximate version of the Ringel-Kotzig conjecture (which asserts the existence of a cyclic packing of 2n-1 copies of any T into the complete graph of order 2n-1) for these trees. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labelling with high probability.

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