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Anti-Ramsey Number of Triangles in Complete Multipartite Graphs

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Abstract

An edge-colored graph is called rainbow if all its edges are colored distinct. The anti-Ramsey number of a graph family \({\mathcal {F}}\) in the graph G, denoted by \(AR{(G,{\mathcal {F}})}\), is the maximum number of colors in an edge-coloring of G without rainbow subgraph in \({\mathcal {F}}\). The anti-Ramsey number for the short cycle \(C_3\) has been determined in a few graphs. Its anti-Ramsey number in the complete graph can be easily obtained from the lexical edge-coloring. Gorgol considered the problem in complete split graphs which contains complete graphs as a subclass. In this paper, we study the problem in the complete multipartite graph which further enlarges the family of complete split graphs. The anti-Ramsey numbers for \(C_3\) and \(C_3^{+}\) in complete multipartite graphs are determined. These results contain the known results for \(C_3\) and \(C_3^{+}\) in complete and complete split graphs as corollaries.

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Acknowledgements

Jin was supported by National Natural Science Foundation of China (11571320 and 12071440) and Zhejiang Provincial Natural Science Foundation (LY19A010018). Sun was supported by Zhejiang Provincial Natural Science Foundation (LY20A010013).

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Authors and Affiliations

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

    Zemin Jin & Kangyun Zhong

  2. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Yuefang Sun

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  1. Zemin Jin
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  2. Kangyun Zhong
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  3. Yuefang Sun
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Correspondence to Zemin Jin.

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Jin, Z., Zhong, K. & Sun, Y. Anti-Ramsey Number of Triangles in Complete Multipartite Graphs. Graphs and Combinatorics 37, 1025–1044 (2021). https://doi.org/10.1007/s00373-021-02302-z

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