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Almost all trees are almost graceful

Adamaszek, Anna, Allen, Peter, Grosu, Codrut and Hladky, Jan (2020) Almost all trees are almost graceful. Random Structures and Algorithms, 56 (4). pp. 948-987. ISSN 1042-9832

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Identification Number: 10.1002/rsa.20906

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 (Formula presented.)), 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.

Item Type: Article
Official URL: https://onlinelibrary.wiley.com/journal/10982418
Additional Information: © 2020 Wiley Periodicals, Inc.
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited: 17 Oct 2019 07:12
Last Modified: 28 Jun 2020 23:27
URI: http://eprints.lse.ac.uk/id/eprint/102133

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