On the anti-Ramsey threshold for non-balanced graphs

TL;DR

This paper develops a framework to identify a broader class of graphs for which the anti-Ramsey threshold in random graphs is smaller than previously established bounds, extending known results and including new examples.

Contribution

It introduces a new framework that encompasses all previously known graphs with smaller anti-Ramsey thresholds and reveals a richer family of such graphs.

Findings

Framework includes all previously known examples

Identifies a larger family of graphs with smaller thresholds

Extends understanding of anti-Ramsey thresholds in random graphs

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.