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Extremal coloring for the anti-Ramsey problem of matchings in complete graphs

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Abstract

Given a graph G, the anti-Ramsey number \(AR(K_n,G)\) is defined to be the maximum number of colors in an edge-coloring of \(K_n\) which does not contain any rainbow G (i.e., all the edges of G have distinct colors). The anti-Ramsey number was introduced by Erdős et al. (Infinite and finite sets, pp 657–665, 1973) and so far it has been determined for several special graph classes. Another related interesting problem posed by Erdős et al. is the uniqueness of the extremal coloring for the anti-Ramsey number. Contrary to the anti-Ramsey number, there are few results about the extremal coloring. In this paper, we show the uniqueness of such extremal coloring for the anti-Ramsey number of matchings in the complete graph.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (11571320, 11671366 and 11401389) and Zhejiang Provincial Natural Science Foundation (LY14A010009, LY15A010008 and LY17A010017). Sun was partially supported by China Scholarship Council (No.201608330111). The authors are very grateful to the referees for helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Zemin Jin, Sherry H. F. Yan & Yuping Zang

  2. Department of Mathematics, Shaoxing University, Shaoxing, 312000, People’s Republic of China

    Yuefang Sun

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  1. Zemin Jin
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  2. Yuefang Sun
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  3. Sherry H. F. Yan
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  4. Yuping Zang
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Correspondence to Sherry H. F. Yan.

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Jin, Z., Sun, Y., Yan, S.H.F. et al. Extremal coloring for the anti-Ramsey problem of matchings in complete graphs. J Comb Optim 34, 1012–1028 (2017). https://doi.org/10.1007/s10878-017-0125-1

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  • DOI: https://doi.org/10.1007/s10878-017-0125-1

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