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Almost all trees are almost graceful

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    0524449 - MÚ 2021 RIV US eng J - Článek v odborném periodiku
    Adamaszek, A. - Allen, P. - Grosu, C. - Hladký, Jan
    Almost all trees are almost graceful.
    Random Structures and Algorithms. Roč. 56, č. 4 (2020), s. 948-987. ISSN 1042-9832
    GRANT EU: European Commission(XE) 628974 - PAECIDM
    Institucionální podpora: RVO:67985840
    Klíčová slova: extremal graph theory * graceful tree labelling * tree
    Kód oboru RIV: BA - Obecná matematika
    Obor OECD: Pure mathematics
    Impakt faktor: 1.008, rok: 2018

    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>n0(γ). Suppose that (i) the maximum degree of T is bounded by Oγ𝛾(n∕log n), 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.
    Trvalý link: http://hdl.handle.net/11104/0308811
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