English

Maximum number of edge colorings avoiding rainbow copies of $K_4$

Combinatorics 2025-05-02 v2

Abstract

In this paper we show that for r12r\geq 12 and any sufficiently large nn-vertex graph GG the number of rr-edge-colorings of GG with no rainbow K4K_4 is at most rex(n,K4)r^{ex(n,K_4)}, where ex(n,K4)ex(n,K_4) denotes the Tur\'{a}n number of K4K_4. Moreover, GG attains equality if and only if it is the Tur\'{a}n graph T3(n)T_3(n). The bound on the number of colors r12r\geq 12 is best possible. It improves upon a result of H. Lefmann, D.A. Nolibos, and the second author who showed the same result for r5434r \geq 5434 and it confirms a conjecture by Gupta, Pehova, Powierski and Staden.

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Cite

@article{arxiv.2503.19244,
  title  = {Maximum number of edge colorings avoiding rainbow copies of $K_4$},
  author = {Hiêp Hàn and Carlos Hoppen and Nicolas Moro Müller and Dionatan Ricardo Schmidt},
  journal= {arXiv preprint arXiv:2503.19244},
  year   = {2025}
}

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15 pages