Prove that the number of orientations of a connected $3$-regular graph on $2n$ vertices where the number of vertices with indegree $0$ and outdegree $0$ are equal, is exactly $2^{n+1}$ $ {2n} \choose {n}$.
Problem
Source: 2020 Dürer Math Competition Finals E+ 1. 5
Tags: combinatorics, graph theory, graph