The generalised rainbow Tur\'an problem for cycles

TL;DR

This paper investigates the maximum number of specific subgraphs in edge-coloured graphs without rainbow cycles, providing precise asymptotic bounds and answering open questions in rainbow Turán problems.

Contribution

It determines the order of magnitude of extremal functions for rainbow cycles and paths in properly edge-coloured graphs, extending known results and resolving prior open questions.

Findings

x(n,C_s,rainbow-C_t) is  for all s,t with sa0a4a0 3.

x(n,C_{2k},rainbow-C_{2k})= for ka0a4a0 3, and x(n,C_{4},rainbow-C_{4})=.

x(n,P_,rainbow-C_{2k}) is  for all k,,  .

Abstract

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number of copies of $H$ that a properly edge-coloured graph on $n$ vertices can contain if it has no rainbow subgraph isomorphic to $F$. We determine the order of magnitude of $\operatorname{ex}(n,C_s,$rainbow-$C_t)$ for all $s,t$ with $s\not =3$. In particular, we answer a question of Gerbner, M\'esz\'aros, Methuku and Palmer by showing that $\operatorname{ex}(n,C_{2k},$rainbow-$C_{2k})$ is $\Theta(n^{k-1})$ if $k\geq 3$ and $\Theta(n^2)$ if $k=2$. We also determine the order of magnitude of $\operatorname{ex}(n,P_\ell,$rainbow-$C_{2k})$ for all $k,\ell\geq 2$, where $P_\ell$ denotes the path with $\ell$ edges.