How many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected?

TL;DR

This paper investigates the conditions under which a dense, randomly edge-colored graph contains multiple edge-disjoint rainbow Hamilton cycles and when it becomes rainbow connected, focusing on the number of colors and edges needed.

Contribution

It introduces new thresholds for the existence of rainbow Hamilton cycles and rainbow connectivity in randomly edge-colored dense graphs.

Findings

Identifies minimum number of colors for rainbow Hamilton cycles

Establishes conditions for rainbow connectivity in dense graphs

Provides probabilistic bounds for rainbow structures in random graphs

Abstract

In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge disjoint rainbow Hamilton cycles. We also ask when in the resultant graph every pair of vertices is connected by a rainbow path.