Mon, 19 Nov 2018
I think I forgot to mention that I was looking recently at hamiltonian cycles on a dodecahedron. The dodecahedron has 30 edges and 20 vertices, so a hamiltonian path contains 20 edges and omits 10. It turns out that it is possible to color the edges of the dodecahedron in three colors so that:
(In this presentation, I have taken one of the vertices and sent it away to infinity. The three edges with arrowheads are all attached to that vertex, off at infinity, and the three faces incident to it have been stretched out to cover the rest of the plane.)
Every face has five edges and there are only three colors, so the colors can't be distributed evenly around a face. Each face is surrounded by two edges of one color, two of a second color, and only one of the last color. This naturally divides the 12 faces into three classes, depending on which color is assigned to only one edge of that face.
Each class contains two pairs of two adjacent pentagons, and each adjacent pair is adjacent to the four pairs in the other classes but not to the other pair in its own class.
Each pair shares a single edge, which we might call its “hinge”, shown here as dotted lines. Each pair has 8 vertices, of which two are on its hinge, four are adjacent to the hinge, and two are not near of the hinge. These last two vertices are always part of the hinges of the pairs of a different class.
I could think about this for a long time, and probably will. There is plenty more to be seen, but I think there is something else I was supposed to be doing today, let me think…. Oh yes! My “job”! So I will leave you to go on from here on your own.
[ Addendum 20181203: David Eppstein has written a much longer and more detailed article about triply-Hamiltonian edge colorings, using this example as a jumping-off point. ]