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    Calculating Ramsey numbers by partitioning colored graphs

    Pokrovskiy, Alexey (2016) Calculating Ramsey numbers by partitioning colored graphs. Journal of Graph Theory 84 (4), pp. 477-500. ISSN 0364-9024.

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    Abstract

    In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring is connected in red, we prove a stronger result - that it is possible to cover all the vertices with k red paths and a blue balanced complete (k+2)-partite graph. Using these results we determine the Ramsey number of a path on n vertices, versus a balanced complete k-partite graph, with m vertices in each part, whenever m-1 is divisible by n-1. This generalizes a result of Erdos who proved the m=1 case of this result. We also determine the Ramsey number of a path on n vertices versus the power of a path on n vertices. This solves a conjecture of Allen, Brightwell, and Skokan.

    Metadata

    Item Type: Article
    Additional Information: This is the peer reviewed version of the article, which has been published in final form at the link above. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.
    School: Birkbeck Schools and Departments > School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Alexey Pokrovskiy
    Date Deposited: 21 Jan 2019 11:13
    Last Modified: 27 Jul 2019 21:53
    URI: http://eprints.bbk.ac.uk/id/eprint/25894

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