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Visualize Hamiltonian Cycles

Make a loop by visiting each vertex only once.

    
Wolfram Language code: dodecahedron = PolyhedronData["Dodecahedron", "Skeleton"];
Find a Hamiltonian cycle:
Wolfram Language code: hcycle = First[FindHamiltonianCycle[dodecahedron]]
Visualize a cycle:
Wolfram Language code: Animate[HighlightGraph[dodecahedron, PathGraph[hcycle[[1 ;; n]]], GraphHighlightStyle -> "Thick"], {n, 0, Length[hcycle], 1}]