Complementary Graphs and the Chromatic Number

Abstract

Zykov proved that if $G$ and $\overline G$ are complementary graphs having chromatic numbers $\chi$ and $\overline \chi$, respectively then $\chi \cdot {\overline \chi}$ is at least the number of vertices of $G$. Nordhaus and Gaddum gave an upper bound for $\chi \cdot {\overline \chi}$ and gave both upper and lower bounds for the analogue $\chi + {\overline \chi}$. In this paper we characterize those graphs for which $\chi \cdot {\overline \chi}$ and $\chi+\overline{\chi}$ reach the bounds of Nordhaus and Gaddum.