Distribution of colours in rainbow H-free colourings

Abstract

An edge colouring of $K_n$ with $k$ colours is a Gallai $k$-colouring if it does not contain any rainbow triangle. Gyárfás, Pálvölgyi, Patkós and Wales proved that there exists a number $g(k)$ such that $n\ge g(k)$ if and only if for any colour distribution sequence $(e_1,\cdots ,e_k)$ with ${\sum }_{i=1}^k{e}_i=\left(\genfrac{}{}{0ex}{}{n}{2}\right)$, there exist a Gallai $k$-colouring of $K_n$ with $e_i$ edges having colour $i$. They also showed that $\Omega (k)=g(k)=O(k^2)$ and posed the problem of determining the exact order of magnitude of $g(k)$. Feffer, Fu and Yan improved both bounds significantly by proving $\Omega (k^{1.5}/\log k)=g(k)=O(k^{1.5})$. We resolve this problem by showing $g(k)=\Theta (k^{1.5}/(\log k{)}^{0.5})$. Moreover, we generalise these definitions by considering rainbow $H$-free colourings of $K_n$ for any general graph $H$, and the natural corresponding quantity $g(H,k)$. We prove that $g(H,k)$ is finite for every $k$ if and only if $H$ is not a forest, and determine the order of $g(H,k)$ when $H$ contains a subgraph with minimum degree at least 3.

15 pages. Submitted to SIAM Journal on Discrete Mathematics