For the tetrahedron, I have:
I tested all Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.
For the tetrahedron, there was simple - there were only four points to study. I obtained more possible arrangements, corresponding to different solutions. The other, not shown is to go on the outside circuit up to the last-but-one point, than to go to the centre and finally at the starting point. There are different solutions only if the start - finish is represented on the figure, otherwise this solution is obtained from the first by a rotation.
For the cube, I worked on an applet:
The strategy I used is the following: I go on the outer lines, up to the last-but-one point. Then I go through the inner one. I can use symmetrical combinations, obtaining more than one path.
Looking at the octahedron, I take the same strategy as for the cube => there are many combinations.
For the dodecahedron, I worked on an applet, taking the same strategy as for the cube, the difference is that here there are more layers.
The last polyhedron I test is the icosahedron:
Here, I worked in the same manner as before, obtaining the figure shown. There are naturally more combinations.