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arXiv · 2201.05106

On the anti-Ramsey threshold for non-balanced graphs

Pedro Araújo·Taísa Martins·Letícia Mattos·Walner Mendonça·Luiz Moreira·Guilherme O. Mota
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Abstract

For graphs $G$ and $H$, we write $G \overset{\mathrm{rb}}{\longrightarrow} H $ 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 $G(n,p) \overset{\mathrm{rb}}{\longrightarrow}H$ 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 and more complex family of such graphs that includes all the previously known examples.

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Pedro Araújo, Taísa Martins, Letícia Mattos, Walner Mendonça, Luiz Moreira, Guilherme O. Mota. 2022-01-13. On the anti-Ramsey threshold for non-balanced graphs. https://arxiv.org/abs/2201.05106

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