On semi-perfect 1-factorizations

The perfect 1-factorization conjecture by A. Kotzig [7] asserts the existence of a 1-factorization of a complete graph K-2n in which any two 1-factors induce a Hamiltonian cycle. This conjecture is one of the prominent open problems in graph theory. Apart from its theoretical significance it has a number of applications, particularly in designing topologies for wireless communication. Recently, a weaker version of this conjecture has been proposed in (1) for the case of semi-perfect 1-factorizations. A semi-perfect 1-factorization is a decomposition of a graph G into distinct 1-factors F-1,...,F-k such that F-1 boolean OR F-i forms a Hamiltonian cycle for any 1 < i <= k. We show that complete graphs K-2n hypercubes Q2(n+1) and tori T-2nx2n admit a semi-perfect 1-factorization.