| Abstract: | Brualdi and Hollingsworth conjectured that, for even n, in a proper edge coloring of  using precisely  colors, the edge set can be partitioned into  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. Krussel, Marshall, and Verrall improved this to three edge disjoint rainbow spanning trees. Recently, Carraher, Hartke and the author proved a theorem improving this to  rainbow spanning trees, even when more general edge colorings of  are considered. In this article, we show that if  is properly edge colored with  colors, a positive fraction of the edges can be covered by edge disjoint rainbow spanning trees. |
