Edge-disjoint rainbow spanning trees in complete graphs

Let G be an edge-colored copy of Kn, where each color appears on at most n/2 edges (the edgecoloring is not necessarily proper). A rainbow spanning tree is a spanning tree of G where each edge has a different color. Brualdi and Hollingsworth [4] conjectured that every properly edge-colored Kn (n ≥ 6 and even) using exactly n−1 colors has n/2 edge-disjoint rainbow spanning trees, and they proved there are at least two edge-disjoint rainbow spanning trees. Kaneko, Kano, and Suzuki [13] strengthened the conjecture to include any proper edge-coloring of Kn, and they proved there are at least three edgedisjoint rainbow spanning trees. Akbari and Alipouri [1] showed that each Kn that is edge-colored such that no color appears more than n/2 times contains at least two rainbow spanning trees. We prove that if n ≥ 1, 000, 000 then an edge-colored Kn, where each color appears on at most n/2 edges, contains at least bn/(1000 logn)c edge-disjoint rainbow spanning trees.