An upper bound for the maximum cut mean value

Abstract

Let MaxCut(G) be the value of the maximum cut of a graph G. Let f(x, n) be the expectation of MaxCut(G)/xn for random graphs with n vertices and xn edges and let r(x,n) be the expectation of MaxCut(G)/xn for random 2x-regular graphs with n vertices. We prove, for sufficiently large x:

$$\lim _{n \to \infty } f(x,n) \leqslant \frac{1}{2} + \sqrt {\frac{{\ln 2}}{{2x}}} ,$$

(1)

$$\lim _{n \to \infty } r(x,n) \leqslant \frac{1}{2} + \frac{1}{{\sqrt x }} + \frac{1}{2}\frac{{\ln x}}{x}.$$

(1)