Almost Every Tree With m Edges Decomposes K 2m, 2m

We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K 2m,2m , a result connected to a conjecture of Graham and Häggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m 2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a random tree have roughly equal size.