  A spanning tree of a properly edge-colored complete graph, $K_n$, is rainbow
provided that each of its edges receives a distinct color. In 1996, Brualdi and
Hollingsworth conjectured that if $K_{2m}$ is properly $(2m-1)$-edge-colored,
then the edges of $K_{2m}$ can be partitioned into $m$ rainbow spanning trees
except when $m=2$. By means of an explicit, constructive approach, in this
paper we construct $\lfloor \sqrt{6m+9}/3 \rfloor$ mutually edge-disjoint
rainbow spanning trees for any positive value of $m$. Not only are the rainbow
trees produced, but also some structure of each rainbow spanning tree is
determined in the process. This improves upon best constructive result to date
in the literature which produces exactly three rainbow trees.
