Rainbow powers of a Hamilton cycle in G(n,p)

TL;DR

This paper determines the threshold probability for the appearance of rainbow powers of Hamilton cycles in randomly edge-colored graphs, showing it closely matches the uncolored case with a slight increase in colors needed.

Contribution

It establishes that the rainbow threshold is within a constant factor of the uncolored threshold, requiring only slightly more colors than the minimum.

Findings

Rainbow Hamilton cycle powers appear at similar thresholds to uncolored cycles.

A small increase in colors suffices to guarantee rainbow structures.

The threshold is within a constant factor of the uncolored case.

Abstract

We show that the threshold for having a rainbow copy of a power of a Hamilton cycle in a randomly edge colored copy of $G_{n,p}$ is within a constant factor of the uncolored threshold. Our proof requires $(1+\varepsilon)$ times the minimum number of colors.