§ Concrete situation:

Let's take a graph $G \equiv (V, E)$. We can imagine that each edge has a potential of $2$. We can redistribute this potential, by providing a potential of $1$ to each of the vertices incident on the edge. This gives us the calculation that the total potential is $2|E|$. But each vertex is assigned a potential of $1$ for each edge incident on it. Thus, the total potential is $\sum_v \texttt{degree}(v)$. This gives the equality $\sum_i \texttt{degree}(v) = 2|E|$.

Thus, if each of the degrees are odd, considering modulo 2, the LHS becomes $\sum_v 1 = |V|$ and the RHS becomes $0$. Thus we have that $|V| = 0$ (mod 2), or the number of vertices remains even.

I learnt of this nice way of thinking about it in terms of potentials when reading a generalization to simplicial complexes.

§ References

• The amazing world of simplicial complexes