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 edge-coloring is not necessarily proper). A rainbow spanning tree is a spanning tree of G where each edge has a different color. Brualdi and Hollingsworth (1996) 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 et al. (2003) strengthened the conjecture to include any proper edge-coloring of Kn, and they proved there are at least three edge-disjoint rainbow spanning trees. Akbari and Alipour (2007) 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 ⌊n/(1000logn)⌋ edge-disjoint rainbow spanning trees.