Nemotron-IMO-Bench 2026 C30

Let $G$ be a graph with $n$ vertices, each of positive degree and let $E$ be the set of edges of $G$. Prove that there are at least $n$ and at most $n!$ of assigning an integer $f(v)$ to each each vertex $v$ such that $\sum_{v} f(v)=|E|$ and for any nonempty subset $A$ of $V$ there is a vertex $v\in A$ such that $f(v)$ is at most the number of neighbors of $v$ in $V\setminus A$.