Nordhaus–Gaddum decompositions for group coloring and DP coloring

Abstract

A Nordhaus–Gaddum theorem states bounds on $p\left(G\right)+p\left(\overline{G}\right)$ and $p\left(G\right)\cdot p\left(\overline{G}\right)$ for some graph parameter $p\left(G\right)$. We consider the sum upper bound for DP coloring and group coloring, and prove that for a graph $G$ with order $n$, ${\chi }_{DP}\left(G\right)+{\chi }_{DP}\left(\overline{G}\right)\le n+1$ and ${\chi }_1\left(G\right)+{\chi }_1\left(\overline{G}\right)\le n+1$. Viewing $\left(G,\overline{G}\right)$ as a decomposition of $K_n$, we determine the extremal decompositions for these bounds.