Edge-disjoint rainbow spanning trees in complete graphs

Let GG be an edge-colored copy of KnKn, where each color appears on at most n/2n/2 edges (the edge-coloring is not necessarily proper). A rainbow spanning tree is a spanning tree of GG where each edge has a different color. Brualdi and Hollingsworth (1996) conjectured that every properly edge-colored KnKn (n≥6n≥6 and even) using exactly n−1n−1 colors has n/2n/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 KnKn, and they proved there are at least three edge-disjoint rainbow spanning trees. Akbari and Alipour (2007) showed that each KnKn that is edge-colored such that no color appears more than n/2n/2 times contains at least two rainbow spanning trees.We prove that if n≥1,000,000n≥1,000,000, then an edge-colored KnKn, where each color appears on at most n/2n/2 edges, contains at least ⌊n/(1000logn)⌋⌊n/(1000logn)⌋ edge-disjoint rainbow spanning trees.