The rainbow vertex-disconnection Nordhaus–Gaddum conjecture

Let $G$ and $\overline{G}$ be nontrivial connected graphs of order $n\geq 8$. Write $rvd(G)$ for the rainbow vertex-disconnection number of $G$. Rainbow vertex-disconnection Nordhaus–Gaddum conjecture.

$$rvd(G)+rvd(\overline{G})\geq n.$$

This conjecture proposes an improvement of the established lower bound for the sum of the rainbow vertex-disconnection numbers of a graph and its complement. The paper notes that the bound holds when $rvd(G)=1$, but does not establish it for all nontrivial connected graphs of order at least $8$.