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Gallai–Ramsey Numbers of Odd Cycles and Complete Bipartite Graphs

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Abstract

For graphs G and H and integer \(k\ge 1\), the Gallai–Ramsey number \(gr_k(G:H)\) is defined to be the minimum integer N such that if \(K_N\) is edge-colored with k colors, then there is either a rainbow G or a monochromatic H. It is known that \(gr_k(K_3:C_{2n+1})\) is exponential in k. In this note, we improve the upper bound for \(gr_k(K_3:C_{2n+1})\) obtained by Hall et al., and give bounds for \(gr_k(K_3:K_{m,n})\).

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Acknowledgements

The authors are grateful to the referees for their invaluable comments, particularly comments for the proof of Lemma 3, which improved the presentation of the manuscript greatly.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Ming Chen, Yusheng Li & Chaoping Pei

  2. College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, Zhejiang, China

    Ming Chen

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  1. Ming Chen
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  2. Yusheng Li
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  3. Chaoping Pei
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Correspondence to Ming Chen.

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This work was supported by NSFC.

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Chen, M., Li, Y. & Pei, C. Gallai–Ramsey Numbers of Odd Cycles and Complete Bipartite Graphs. Graphs and Combinatorics 34, 1185–1196 (2018). https://doi.org/10.1007/s00373-018-1931-7

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  • DOI: https://doi.org/10.1007/s00373-018-1931-7

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