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 ${\frak P}\_2$?



"Nice" and "nasty" partitions in graphs
Such a graph exists if and only if $n \geq 4$.


The inequality defining "nice" implies that each set in your partition has at least two elements. Thus we need $n \geq 4$.


If $n \geq 4$ is even, then let $G$ be $\frac{n}{2}$ disjoint $2$-paths. This idea is explained in Gordon Royle's comment: the partition taking one vertex from each $2$-path is nasty, and the partition putting some $2$-paths in one set and some in another is nice.


If $n \geq 4$ is odd, then let $G$ consist of $\frac{n-1}{2}$ disjoint $2$-paths plus a $3$-path. For a nasty partition, take one vertex from each $2$-path plus the degree-two vertex from the $3$-path. For a nice partition, take the $2$-paths in one set and the $3$-path in another.


