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Almost all trees are almost graceful

Adamaszek, Anna, Allen, Peter, Grosu, Codrut and Hladky, Jan (2019) Almost all trees are almost graceful. Random Structures and Algorithms. ISSN 1042-9832 (In Press)

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Abstract

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 γ > 0 and for all n > n0(γ). Suppose that (i) the maximum degree of T is bounded by Oγ(n/ log n), and (ii) the vertex labels are chosen from the set {1, 2, . . . , d(1 + γ)ne}. 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. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labelling with high probability.

Item Type: Article
Divisions: Mathematics
Date Deposited: 17 Oct 2019 07:12
Last Modified: 13 Nov 2019 00:32
URI: http://eprints.lse.ac.uk/id/eprint/102133

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