Packing trees of unbounded degrees in random graphs

In this paper, we address the problem of packing large trees in G_n,p. In particular, we prove the following result. Suppose that T₁, . . . , T_N are n-vertex trees, each of which has maximum degree at most (np)^1/6/(log n)⁶. Then with high probability, one can find edge-disjoint copies of all the T_i in the random graph Gn,p, provided that p ≥ (log n)36/n and N ≤(1 − ε)np/2 for a positive constant ⊂. Moreover, if each T_i has at most (1 − α)n vertices, for some positive α, then the same result holds under the much weaker assumptions that p ≥(log n)2/(cn) and Δ(Ti) ≤ cnp/ log n that depends only on α and ε. Our assumptions on maximum degrees of the trees are significantly weaker than those in all previously known approximate packing results.