On the anti-Ramsey threshold for non-balanced graphs

For graphs $G$ and $H$, we write $GH$ if any proper edge-coloring of $G$ contains a rainbow copy of $H$, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last author proved that the threshold for $G(n,p) \overset{$\mathrm{r}\mathrm{b}$}{\longrightarrow}H$isatmost$n^{-1/m_2(H)}$. Previous results have matched the lower bound for this anti-Ramsey threshold for cycles and complete graphs with at least 5 vertices. Kohayakawa, Konstadinidis and the last author also presented an infinite family of graphs $H$ for which the anti-Ramsey threshold is asymptotically smaller than $n^{-1\mathrm{/}{m}_2(H)}$. In this paper, we devise a framework that provides a richer and more complex family of such graphs that includes all the previously known examples.