On the anti-Ramsey threshold for non-balanced graphs (Q6131744)

Summary: For graphs \(G\), \(H\), we write \(G \overset{\text{rb}}{\longrightarrow} H\) if for every proper edge-coloring of \(G\) there is a rainbow copy of \(H\), i.e., a copy where no color appears more than once. \textit{Y. Kohayakawa} et al. [J. Graph Theory 87, No. 2, 176--187 (2018; Zbl 1380.05118)] proved that the threshold for \(G(n, p) \overset{\text{rb}}{\longrightarrow} H\) is at most \(n^{-1/m_2(H)}\). Previous results have matched the lower bound for this anti-Ramsey threshold for cycles and complete graphs with at least 5 vertices. Kohayakawa et al. [loc. cit.] also presented an infinite family of graphs \(H\) for which the anti-Ramsey threshold is asymptotically smaller than \(n^{-1/m_2(H)}\). In this paper, we devise a framework that provides a richer family of such graphs.