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Visualize Hamiltonian Cycles 

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X Manipulate[ HighlightGraph[ g2, {Style[hcycle[[1, 1]], Red], PathGraph[hcycle[[1 ;; n]]]}, GraphHighlightStyle -> "Thick", ImageSize -> {400, 300}], {n, 0, Length[hcycle], 1}, Initialization :> (g2 = PolyhedronData["Dodecahedron", "SkeletonGraph"]; hcycle = First[FindHamiltonianCycle[g2]]) ]

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