Almost all trees are almost graceful. Issue 4 (19th February 2020) - Adamaszek, Anna; Allen, Peter; Grosu, Codruţ; Hladký, Jan
Abstract : The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled 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 γ >0 and for all n > n 0 ( γ ). Suppose that (i) the maximum degree of T is bounded by O γ ( n / log n ), and (ii) the vertex labels are chosen from the set 1, 2, …, ⌈(1+ γ ) n ⌉. Then there is an injective labeling 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 labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability.