Construction of hamiltonian cycles with a given spectrum of edge directions in an n-dimensional Boolean cube

Abstract

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 2k. 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.