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arXiv · 2503.19244

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

Hiêp Hàn·Carlos Hoppen·Nicolas Moro Müller·Dionatan Ricardo Schmidt
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Abstract

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

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Hiêp Hàn, Carlos Hoppen, Nicolas Moro Müller, Dionatan Ricardo Schmidt. 2025-03-25. Maximum number of edge colorings avoiding rainbow copies of $K_4$. https://arxiv.org/abs/2503.19244

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