Spanning Trees Orthogonal to One-Factorizations of \(K_{2n}\)

In [3] Brualdi and Hollingsworth conjectured that for any one-factorization \(\mathcal{F}\) of \(K_n\), there exists a decomposition of \(K_{2n}\) into spanning trees orthogonal to \(\mathcal{F}\). They also showed that two such spanning trees always existed. We construct three such trees and exhibit an infinite class of complete graphs with an orthogonal decomposition into spanning trees with respect to the one-factorization \(GK_{2n}\).