Rainbow spanning trees in complete graphs colored by one-factorizations

Brualdi and Hollingsworth conjectured that, for even n, in a proper edge coloring of Kn using precisely n − 1 colors, the edge set can be partitioned into n2 spanning trees which are rainbow (and hence, precisely one edge from each color class is in each spanning tree). They proved that there always are two edge disjoint rainbow spanning trees. Kaneko, Kano and Suzuki improved this to three edge disjoint rainbow spanning trees. Recently, Carraher, Hartke and the author proved a theorem improving this to n logn rainbow spanning trees, even when more general edge colorings of Kn are considered. In this paper, we show that if Kn is properly edge colored with n−1 colors, a positive fraction of the edges can be covered by edge disjoint rainbow spanning trees.