Library Header Image
LSE Research Online LSE Library Services

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)

[img] Text (Almost all trees are almost graceful) - Accepted Version
Pending embargo until 1 January 2100.

Download (638kB) | Request a copy


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: 22 Jan 2020 00:26

Actions (login required)

View Item View Item


Downloads per month over past year

View more statistics