Packing trees into complete bipartite graphs

In 1976, Gyárfás and Lehel conjectured that any trees T 2 , … , T n with 2 through n vertices pack into K n , the complete graph on n vertices, that is, the trees T 2 , … , T n appear as edge-disjoint subgraphs of K n . This conjecture is still unresolved. mine an analogous conjecture for packing trees into complete bipartite graphs. Let T a , a denote a tree whose partite sets both have size a , which we call a balanced tree. We conjecture that any trees T 1 , 1 , … , T n , n pack into K n , n , the complete bipartite graph on 2 n vertices. in by establishing that if a and n are integers with n ≥ 3 and a < ⌊ 7 / 18 n ⌋ , then any balanced trees T 1 , 1 , … , T a , a pack into K n , n . o show that if any degree sequence for the first partite set is specified for each tree, then there exist balanced trees T 1 , 1 , … , T n , n with these vertex degrees that pack into K n , n .