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    Ramsey goodness of paths

    Pokrovskiy, Alexey and Sudakov, B. (2017) Ramsey goodness of paths. Journal of Combinatorial Theory, Series B 122 , pp. 384-390. ISSN 0095-8956.

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    Abstract

    Given a pair of graphs G and H, the Ramsey number R(G,H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G,H)≥(|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)-coloring of H. A graph G is called H-good if R(G,H)=(|G|−1)(χ(H)−1)+σ(H). The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n≥4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan.

    Metadata

    Item Type: Article
    Keyword(s) / Subject(s): Ramsey numbers, Ramsey goodness, Paths in graphs, Expanders
    School: School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Alexey Pokrovskiy
    Date Deposited: 21 Jan 2019 11:15
    Last Modified: 30 Jun 2020 03:08
    URI: https://eprints.bbk.ac.uk/id/eprint/25895

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