On the Tree Packing Conjecture

The Gyárfás tree packing conjecture states that any set of n−1 trees T1, T2, . . . , Tn−1 such that Ti has n− i+ 1 vertices pack into Kn (for n large enough). We show that t = 1 10n 1/4 trees T1, T2, . . . , Tt such that Ti has n− i+ 1 vertices pack into Kn+1 (for n large enough). We also prove that any set of t = 1 10 n1/4 trees T1, T2, . . . , Tt such that no tree is a star and Ti has n− i+ 1 vertices pack into Kn (for n large enough). Finally, we prove that t = 1 4 n1/3 trees T1, T2, . . . , Tt such that Ti has n− i + 1 vertices pack into Kn as long as each tree has maximum degree at least 2n2/3 (for n large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Komlós, Sárközy and Szemerédi [15].