Enumeration of Hamiltonian Cycles in 6-cube

TL;DR

This paper precisely enumerates the total and automorphism classes of directed Hamiltonian cycles in the 6-cube, providing exact counts that were previously estimated or unknown.

Contribution

It presents the first exact enumeration of Hamiltonian cycles in the 6-cube, including counts up to automorphisms, advancing combinatorial understanding of hypercube structures.

Findings

Exact count of Hamiltonian cycles: 14,754,666,508,334,433,250,560

Number of cycles up to automorphisms: 147,365,405,634,413,085

Addresses a problem listed in Knuth's 'The Art of Computer Programming'

Abstract

Finding the number 2H6 of directed Hamiltonian cycles in 6-cube is problem 43 in Section 7.2.1.1 of Knuth's ' The Art of Computer Programming'; various proposed estimates are surveyed below. We computed exact value: H6=14,754,666,508,334,433,250,560=6*2^4*217,199*1,085,989*5,429,923. Also the number Aut6 of those cycles up to automorphisms of 6-cube was computed as 147,365,405,634,413,085