How many random edges make a dense graph hamiltonian?

TL;DR

This paper determines the number of random edges needed to make a dense graph Hamiltonian, showing that adding Theta(n) edges suffices generally, but fewer are needed if the graph lacks large independent sets, with similar results for directed graphs.

Contribution

It establishes tight bounds on the number of random edges needed for Hamiltonicity in dense graphs, including special cases and directed graphs.

Findings

Theta(n) random edges are necessary and sufficient for general dense graphs.

Fewer edges are needed if the graph has no large independent set.

Results extend to directed graphs.

Abstract

This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding $\Theta(n)$ random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs.