How many random edges make a dense graph Hamiltonian?

Authors:
Tom Bohman;Alan Frieze;Ryan Martin
Affiliations:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania;Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania;Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania
Venue:
Random Structures & Algorithms
Year:
2003

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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 Θ(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.