Let $G=(V,E)$ be a finite, simple, undirected graph such that $\bigcup E = V$ (that is, every vertex belongs to at least one edge).

For $v\in V$ we set $N(v) = \{w\in V:\{v,w\}\in E\}$, and for $S\subseteq V$ we let $N(S) = \bigcup \{N(v):v\in S\}$. A *bijective neighborhood map* is a bijection $\varphi:V\to V$ such that for all $v\in V$ we have $\varphi(v) \in N(v)$.

Clearly, a necessary condition for a graph $G=(V,E)$ to have a bijective neighborhood map is that

(C) for all non-empty $S\subseteq V$ we have $|S|\leq |N(S)|$.

(A very similar condition is central to Hall's theorem.)

**Question.** Is (C) also sufficient for a graph $G$ to have a bijective neighborhood function?