Nordhaus-Gaddum-Type Results

Abstract

Let $\mathcal{G}(n)$ denote the class of simple graphs of order n and $\mathcal{G}(n,m)$ the subclass of $\mathcal{G}(n)$ having graphs with n vertices and m edges. Given a graph parameter f(G) and a positive integer n, the Nordhaus-Gaddum Problem is to determine sharp bounds for (1) $f(G) + f(\overline{G})$ and (2) $f(G) \cdot f(\overline{G})$, as G ranges over the class $\mathcal{G}(n)$, and characterize the extremal graphs. The Nordhaus-Gaddum-type relations have received wide attention; see a survey paper [3] by Aouchiche and Hansen.