Dear,

First, I managed to sort properly the unlabeled necklaces on a torus (UN, 6 beads, 6 colors, there are 43 such necklaces)

http://theory.cs.uvic.ca/inf/neck/NecklaceInfo.html) .

000000

000001

.......11

.......12

..........

012345

The post includes a question about graph and Hamiltonian cycles. Alas, it 's not yet a standard graph, because some vertexes of the considered graph must be erased while propagating within the graph. This is because the nominal nodes include all the representatives of given classes (i.e. the 6 rotations of each Unlabelled Necklace) while the cyclic path of interest must includes a single representative of each class. (the graph is defined of the set of the 43x6 (rotation) necklaces (vertex) and edges connect the 43x6 necklaces that only differ by the bead at position 0)

I foresee to defend that the proposed classifying of the necklaces follows the most basic possible rule that defines the torus : only one beads changes from a necklace to its neighbor on the torus and this bead is always at the same place.

The rule is the following, you start from a given necklace UN(0) and a given rotation of UN(0), the next necklace on the torus is UN(i+1)=rotate(Un,1 bead clockwise) + change bead(0)

A physical model is derived from the density of state of the UN in images of natural scenes.

This torus offers the possibility to define a law group (i.e. the law group on the nth roots of 1) on the set of the unlabelled necklaces and it opens a way to rule a physical model on shapes, a basis of shapes are the UN. The necklace 000000, that only have a single neighbor (i.e. 000001), can not be included into the torus, has been extracted from the list ; it is used to define a zero for a complex plane in the UN space while the torus is the equivalent of the unit circle in the complex plane.

Billions of pathes have been tested yet and a single hamiltonian path on the UN has been carried out. The combinatorics to define a torus is not tractable and because the graph is changing during path exploration, there is no way yet to use the standard algorithms that help to find hamiltonian cycles.

The question is the following : is there a bibliography on graphes where vertexes may change during propagation of a path? I found nothing about this topics on graduate and post-graduate courses and I lost my brain on Reutenauer's papers (who on Earth was that so genius guy?), Hopf algebra and alike. I envisioned that the proposed graph may be extended to a standard graph (with stated vertex/nodes) ...

Thank you