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 K2n 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 F1, . . . , Fk such that F1 ∪ Fi forms a Hamiltonian cycle for any 1 < i ≤ k. We show that complete graphs K2n, hypercubes Q2n+1 and tori T2n×2n admit a semi-perfect 1-factorization.