Distribution of colours in rainbow H-free colourings

TL;DR

This paper determines the precise growth rate of the function g(k) related to rainbow triangle-free colourings of complete graphs, and generalizes the concept to other graphs H, establishing conditions for finiteness and growth of g(H,k).

Contribution

It exactly characterizes the asymptotic behavior of g(k) and extends the framework to rainbow H-free colourings, identifying when g(H,k) is finite and its order of magnitude.

Findings

g(k) = Θ(k^{1.5}/(log k)^{0.5})

g(H,k) is finite iff H is not a forest

g(H,k) has specific growth when H contains a subgraph with minimum degree at least 3

Abstract

An edge colouring of $K_n$ with $k$ colours is a Gallai $k$-colouring if it does not contain any rainbow triangle. Gy\'arf\'as, P\'alv\"olgyi, Patk\'os and Wales proved that there exists a number $g(k)$ such that $n\geq g(k)$ if and only if for any colour distribution sequence $(e_1,\cdots,e_k)$ with $\sum_{i=1}^ke_i=\binom{n}{2}$, 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.