Anti-Ramsey threshold of cycles

TL;DR

This paper determines the probability threshold at which a random graph almost surely contains a rainbow cycle of a given length under any proper edge colouring.

Contribution

It extends previous work to establish the exact threshold for rainbow cycles of any length in random graphs.

Findings

Threshold for rainbow cycles in G(n,p) determined for all cycle lengths.

Generalizes previous results to cycles of length ≥ 4.

Provides a precise probabilistic boundary for rainbow cycle existence.

Abstract

For graphs $G$ and $H$, let $G \overset{\mathrm{rb}}{{\longrightarrow}} H$ denote the property that for every proper edge colouring of $G$ there is a rainbow copy of $H$ in $G$. Extending a result of Nenadov, Person, \v{S}kori\'{c} and Steger [J. Combin. Theory Ser. B 124 (2017),1-38], we determine the threshold for $G(n,p) \overset{\mathrm{rb}}{{\longrightarrow}} C_\ell$ for cycles $C_\ell$ of any given length $\ell \geq 4$.