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If an edge is fixed and is directed, how many hamiltonian paths will there be in a cube or an 8-vertexed graph? 6 vertices are not yet traversed. And how many of them will be hamiltonian cycles?

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  • $\begingroup$ Do you know how many Hamiltonian paths (or, cycles) there are in a cube, without the restriction? Since all the edges are equivalent, it shouldn't be hard to work out how many of the paths (or, cycles) contain any one particular directed edge. $\endgroup$
    – Gerry Myerson
    Commented Oct 13 at 1:04

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