On the Anti-Ramsey Threshold for Non-Balanced Graphs

• Published in: The Electronic journal of combinatorics Vol. 31; no. 1
• Main Authors: Araújo, Pedro, Martins, Taísa, Mattos, Letícia, Mendonça, Walner, Moreira, Luiz, Mota, Guilherme O.
• Format: Journal Article
• Language: English
• Published: 22-03-2024
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/11449

For graphs $G,H$, we write $G \overset{\mathrm{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. Kohayakawa, Konstadinidis and the last author proved that the threshold for $G(n,p) \overset{\mathrm{rb}}{\longrightarrow} H$ 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, Konstadinidis and the last author also presented an infinite family of graphs $H$ for which the anti-Ramsey threshold is asymptotically smaller than $n^{-1/m_2(H)}$. In this paper, we devise a framework that provides a richer family of such graphs.