Let $G=(V,E)$ be a graph. For $v\in V$ we set $N(v) = \{w\in V: \{v,w\}\in E\}$. Let us say that a graph is neighborly if there is an injective function $f: V\to V$ such that $f(v)\in N(v)$ for all $v\in V$. (Is there standard terminology for this? Any pointers appreciated!)
If a graph has a Hamiltonian cycle, let's call it a HC-graph. It is easy to see that HC-graphs are neighborly, as well as unions of HC-graphs, each of which has more than 1 vertex.
Is the following statement correct?
(?): If a finite graph $G=(V,E)$ is neighborly, then there is a partition of $V$ into subgraphs, each of which is a HC-graph and consists of more than 1 point.
(Remark: The nice thing about this definition is that there are neighborly graphs for any finite or infinite cardinality, but it is not obvious how to generalise Hamiltonian cycles to infinite graphs.)