Construction of Hamiltonian Cycles with a Given Spectrum of Edge Directions in an n-Dimensional Boolean Cube

The spectrum of a Hamiltonian cycle (of a Gray code) in an n-dimensional Boolean cube is the series a = (a1, . . . , an), where ai is the number of edges of the ith direction in the cycle. The necessary conditions for the existence of a Gray code with the spectrum a are available: the numbers ai are even and, for k = 1, . . . , n, the sum of k arbitrary components of a is at least 2. We prove that there is some dimension N such that if the necessary condition on the spectrum is also sufficient for the existence of a Hamiltonian cycle with the spectrum in an N-dimensional Boolean cube then the conditions are sufficient for all dimensions n. DOI: 10.1134/S1990478912030088