Low dimensional middle cubes are Hamiltonian

Abstract

Let Q _n be the n-dimensional hypercube. If n=2k+1, then the subgraph M _n of Q _n induced by the nodes having exactly k or k + 1 ones is called the middle cube of dimension n. A famous conjecture is that M _n is hamiltonian if n>1.

Clearly, M ₃ has two oriented hamiltonian cycles. This paper includes the following new results:

• 1. The number of oriented hamiltonian cycles of M _n is divisible by k!(k + 1)!.

• 2. M₅ has exactly 48 oriented hamiltonian cycles.

• 3. M₇ is hamiltonian.

• 4. Some interesting new combinatorial identities which are relevant to the structure of M _n.

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