On the problem of a matching orthogonal to a 2-factorization

A 2-factorization of a 2d-regular graph G has an orthogonal matching if there is a matching of G in which each edge lies in a distinct 2-factor. We show that there is always an orthogonal subgraph in which each component is either a single edge or a path of length two and the number of paths of length two is at most min{25d,max{415d+815,2d}}.