Nordhaus–Gaddum inequality

Statement

Let G be a graph with n vertices such that $G\neq K_n, \overline{K_n}$ where $\overline{G}$ denotes the complement of G. Then $n + 2 \leq \chi_i(G) + \chi_{i}(\overline{G})\leq 2n-1.$ These bounds are sharp for all values of n.