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

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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 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 |
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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: | 20 Jul 2021 02:50 |

URI: | http://eprints.lse.ac.uk/id/eprint/102133 |

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