On the existence of a matching orthogonal to a 2-factorization (Q1262879)

Brian Alspach in 1988 posed the following problem: For any 2- factorization \(F_ 1,F_ 2,...,F_ d\) of a 2d-regular graph G, is it possible to find a d-matching M in G such that M contains precisely one edge from each of \(F_ 1,F_ 2,...,F_ d?\) The authors of this article show that if \(F_ 1,F_ 2,...,F_ d\) is a 2-factorization of a 2d- regular graph G of order \(n\geq 3,23d\), then G contains a d-matching with exactly one edge from each of \(F_ 1,F_ 2,...,F_ d\).