The longest distance travelled by an ant on the sides of a cube.

This is a problem from IWYMIC-Individual $1999$:

Section B. Answer the following 3 questions, and show your detailed solution in the space provided after each question.

The diagram below shows a cubical wire framework of side $1$. An ant starts from a vertex and crawls along the sides of the framework. If it does not repeat any part of its path and finally returns to the starting vertex, what is the longest possible length of the path it has travelled?

$\qquad\qquad\qquad\qquad\qquad$

Just trying different ways, I realized it can be done by moving along $8$ sides of the cube as follows (and the answer sheets agree with it):

$\qquad\qquad\qquad\qquad\qquad\qquad$

However, the problem requires a detailed solution. The perimeter of the cube is $12$ and I believe it's enough to show that the path length cannot be longer than $8$, but I'm not sure how to do it. I think maybe using the cube's net can help.

There are only 8 vertices. Suppose a path was 9 (or more) edges long. Each edge of that path has an ending vertex, so you had to enter at least one vertex twice ...

Hint: Notice that the number of edges that the number of times you enter a vertex must be equal to the number of times you leave it, so in particular, each vertex is connected to an even number of edges of your path (possibly $0$) and find a relationship between the number of times it passes through a vertex and the number of edges it walks through