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)

Text (Almost all trees are almost graceful)
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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 |
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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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