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Ramsey Numbers of Multiple Copies of Graphs in a Component

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Abstract

For a graph G, let \(R({\mathcal {C}}(nG))\) denote the least N such that every 2-colouring of the edges of \(K_N\) contains a monochromatic copy of nG in a monochromatic connected subgraph, where nG denotes n vertex disjoint copies of G. Gyárfás and Sárközy (J Graph Theory 83(2):109–119, 2016) showed that \(R({\mathcal {C}}(nK_3))=7n-2\) for \(n \ge 2\). After that, Roberts (Electron J Comb 24(1):8, 2017)showed that \(R({\mathcal {C}}(nK_r))=(r^2-r+1)n-r+1\) for \(r \ge 4\) and \(n \ge R(K_r)\), where \(R(K_r)\) is the Ramsey number of \(K_r\). In this paper, we determine \(R({\mathcal {C}}(nG))\) for all 4-vertex graphs G without isolated vertices.

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Acknowledgements

We are thankful to the reviewers for reading the manuscript carefully and giving very valuable comments to help improve the presentation. This research is supported by National Natural Science foundation of China (Grant Nos. 11931002 and 12371327).

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The research was partially supported by NSFC (Grant Nos. 11931002 and 12371327).

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  1. School of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China

    Caixia Huang, Yuejian Peng & Yiran Zhang

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  1. Caixia Huang
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  2. Yuejian Peng
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Correspondence to Yuejian Peng.

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Huang, C., Peng, Y. & Zhang, Y. Ramsey Numbers of Multiple Copies of Graphs in a Component. Graphs and Combinatorics 40, 94 (2024). https://doi.org/10.1007/s00373-024-02821-5

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