Let $G=(V,E)$ be a simple, undirected graph, that is $V$ is a set and $E \subseteq [V]^2 = \{\{v,w\}: v,w \in V \land v\neq w\}$.

For $v\in V$ and $S\subseteq V$ we set $$N(v,S) = \{w\in S: \{v,w\} \in E\}.$$ Given a partition $\frak P$ of $V$ into $2$ non-empty sets, and a vertex $v\in V$, we denote by $[v]$ the unique member of $\frak P$ containing $v$, and by $\neg[v]$ the unique member of $\frak P$ *not* containing $v$.

A partition $\frak P$ of $V(G)$ into 2 sets is said to be *nice* if $N(v,[v])> N(v,\neg[v])$ for all $v\in V$, and we call it *nasty* if $N(v,[v])< N(v,\neg[v])$ for all $v\in V$.

For which $n\in\mathbb{N}$ is there a graph $G$ on $n$ vertices such that $G$ has both a nice partition ${\frak P}_1$ as well as a nasty partition ${\frak P}_2$?