Abstract
Given two graphs G and H, the general k-colored Gallai–Ramsey number \({\text {gr}}_k(G:H)\) is defined to be the minimum integer m such that every k-coloring of the complete graph on m vertices contains either a rainbow copy of G or a monochromatic copy of H. Interesting problems arise when one asks how many such rainbow copy of G and monochromatic copy of H must occur. The Gallai–Ramsey multiplicity \({\text {GM}}_{k}(G:H)\) is defined as the minimum total number of rainbow copy of G and monochromatic copy of H in any exact k-coloring of \(K_{{\text {gr}}_{k}(G:H)}\). In this paper, we give upper and lower bounds for Gallai–Ramsey multiplicity involving some small rainbow subgraphs.
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The author is very grateful to the referees for their valuable comments and suggestions.
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Supported by the National Science Foundation of China (No. 12061059) and the Qinghai Key Laboratory of Internet of Things Project (2017-ZJ-Y21).
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Mao, Y. Gallai–Ramsey Multiplicity. Graphs and Combinatorics 40, 54 (2024). https://doi.org/10.1007/s00373-024-02780-x
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DOI: https://doi.org/10.1007/s00373-024-02780-x