Multicolored forests in complete bipartite graphs

Abstract If the edges of a graph G are colored using k colors, we consider the color distribution for this coloring a =(a 1 ,a 2 ,…,a k ) , in which ai denotes the number of edges of color i for i=1,2,…,k. We find inequalities and majorization conditions on color distributions of the complete bipartite graph Kn,n which guarantee the existence of multicolored subgraphs: in particular, multicolored forests and trees. We end with a conjecture on partitions of Kn,n into multicolored trees.