Rainbow Hamilton cycles in random graphs

TL;DR

This paper proves that in a random graph with edges colored from about n colors, a rainbow Hamilton cycle almost always exists near the known threshold for Hamiltonicity.

Contribution

It establishes the first-order threshold for the existence of rainbow Hamilton cycles in randomly edge-colored graphs, extending classical Hamiltonicity results.

Findings

Rainbow Hamilton cycles appear with high probability when p ~ (log n)/n.

Achieves the best possible first order constant for the threshold.

Connects edge-colored graphs to hypergraph Hamiltonicity theory.

Abstract

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdos-Renyi random graph G_{n,p} is around p ~ (log n + log log n) / n. Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform hypergraph by connecting 3-uniform hypergraphs to edge-colored graphs. In this work, we consider that setting of edge-colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of G_{n,p} are randomly colored from a set of (1 + o(1)) n colors, with p = (1 + o(1)) (log n) / n, we show that one can almost always find a Hamilton cycle which has the further property that all edges are distinctly colored (rainbow).