A probabilistic result for the max-cut problem on random graphs

We consider the max-cut problem on a random graph G with n vertices and weights w(ij) being independent bounded random variables with the same fixed positive expectation mu and variance sigma (2). It is well known that the max-cut number mc(G) always exceeds 1/2 Sigma (i <j)w(ij). We prove that with probability greater than p(n), the max-cut number satisfies 1/2 Sigma (i <j)w(ij)less than or equal to mc(G)less than or equal toq(n)(1/2 Sigma (i <j)w(ij)). where p(n),q(n) are explicitly expressed in terms of the problem's data and such that p(n),q(n) approach 1 as n --> infinity. (C) 2000 Elsevier Science B.V. All rights reserved.