How many random edges make a dense graph hamiltonian?

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.

Embedding spanning bounded degree subgraphs in randomly perturbed graphs

“…We now change the setup in the following way, as first suggested by Bohman, Frieze and Martin [6] (though they worked with G(n, m) instead of G(n, p)). For α ∈ (0, 1), let G α be any graph with minimum degree at least αn and reveal more edges within the graph independently at random with probability p. That is, we study the properties of G α ∪ G(n, p).…”

“…For α ∈ (0, 1/2) Bohman, Frieze and Martin [6] showed that if p = ω(1/n) then whp there is a Hamilton cycle in G α ∪ G(n, p) for any G α . Furthermore, this is optimal, as for p = o(1/n) there are graphs G α such that G α ∪ G(n, p) is not Hamiltonian whp.…”

On two Hamilton cycle problems in random graphs

“…For example, Bohman, Frieze and Martin [2] considered graphs of the form G = H + R where H is arbitrary, but with high minimum degree and R is random. In this section we consider graphs of the form G = R − H where R is random and H is an arbitrary subset of R, subject to some restrictions.…”

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

“…The random graph model was first introduced by Bohman, Frieze, and Martin . It can be considered as an extension of the Erdős‐Rényi model, which we can retrieve from by setting .…”

“…How many are needed so that the result is a graph that satisfies property w.h.p . In , they study the case where is the set of graphs of minimum degree and is the property of a graph having a Hamilton cycle. They show that a linear number of random edges suffices to make any member of Hamiltonian w.h.p.…”