Low dimensional middle cubes are Hamiltonian
Let Qn be the n-dimensional hypercube. If n=2k+1, then the subgraph Mn of Qn induced by the nodes having exactly k or k + 1 ones is called the middle cube of dimension n. A famous conjecture is that Mn is hamiltonian if n>1.
The number of oriented hamiltonian cycles of Mn is divisible by k!(k + 1)!.
M5 has exactly 48 oriented hamiltonian cycles.
M7 is hamiltonian.
Some interesting new combinatorial identities which are relevant to the structure of Mn.
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