A066037 Number of (undirected) Hamiltonian cycles in the binary n-cube, or the number of cyclic n-bit Gray codes.

1, 1, 6, 1344, 906545760, 35838213722570883870720

Offset: 1

Author

John Tromp, Dec 12 2001

Keywords

• nonn
• nice
• more

Comments

This is the number of ways of making a list of the 2^n nodes of the n-cube, with a distinguished starting position and a direction, such that each node is adjacent to the previous one and the last node is adjacent to the first; and then dividing the total by 2^(n+1) because the starting node and the direction do not really matter.

The number is a multiple of n!/2 since any directed cycle starting from 0^n induces a permutation on the n bits, namely the order in which they first get set to 1.

Examples

			The 2-cube has a single cycle consisting of all 4 edges.
		

Links

• Michel Deza and Roman Shklyar, Enumeration of Hamiltonian Cycles in 6-cube, arXiv:1003.4391 [cs.DM], 2010. [There may be errors - see Haanpaa and Ostergard, 2012]
• R. J. Douglas, Bounds on the number of Hamiltonian circuits in the n-cube, Discrete Mathematics, 17 (1977), 143-146.
• Harri Haanpaa and Patric R. J. Östergård, Counting Hamiltonian cycles in bipartite graphs, Math. Comp. 83 (2014), 979-995.
• Harary, Hayes, and Wu, A survey of the theory of hypercube graphs, Computers and Mathematics with Applications, 15(4), 1988, 277-289.
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle
• Eric Weisstein's World of Mathematics, Hypercube Graph

Crossrefs

Equals A006069/2^(n+1) and A003042/2.

Cf. A006070, A091299, A003043, A091302.

Cf. A236602 (superset). - Stanislav Sykora, Feb 01 2014

Programs

Mathematica

Prepend[Table[Length[FindHamiltonianCycle[HypercubeGraph[n], All]], {n, 2, 4}], 1] (* Eric W. Weisstein, Apr 01 2017 *)

Extensions

a(6) from Michel Deza, Mar 28 2010

a(6) corrected by Haanpaa and Östergård, 2012. - N. J. A. Sloane, Sep 06 2012

Name clarified by Eric W. Weisstein, May 06 2019