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The Ramsey numbers of squares of paths and cycles

Allen, Peter ORCID: 0000-0001-6555-3501, Mergoni Cecchelli, Domenico, Skokan, Jozef ORCID: 0000-0003-3996-7676 and Roberts, Barnaby (2024) The Ramsey numbers of squares of paths and cycles. Electronic Journal of Combinatorics, 31 (2). ISSN 1077-8926

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Identification Number: 10.37236/11847

Abstract

The square G 2 of a graph G is the graph on V (G) with a pair of vertices uv an edge whenever u and v have distance 1 or 2 in G. Given graphs G and H, the Ramsey number R(G, H) is the minimum N such that whenever the edges of the complete graph K N are coloured with red and blue, there exists either a red copy of G or a blue copy of H. We prove that for all sufficiently large n we have (Formula presented). We also show that for every γ > 0 and ∆ there exists β > 0 such that the following holds: If G can be coloured with three colours such that all colour classes have size at most n, the maximum degree of G is at most ∆, and G has bandwidth at most βn, then R(G, G) ≤ (3 + γ)n.

Item Type: Article
Official URL: https://www.combinatorics.org/ojs/index.php/eljc
Additional Information: © 2024 The Author(s)
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 26 Mar 2024 15:09
Last Modified: 11 Jun 2024 23:36
URI: http://eprints.lse.ac.uk/id/eprint/122505

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