Packing trees of unbounded degrees in random graphs

Asaf Ferber, Wojciech Samotij

Published 2016-07-25Version 1

In this paper, we address the problem of packing large trees in $G_{n,p}$. In particular, we prove the following result. Suppose that $T_1, \dotsc, T_N$ 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 $G_{n,p}$, provided that $p \geq (\log n)^{36}/n$ and $N \le (1-\varepsilon)np/2$ for a positive constant $\varepsilon$. Moreover, if each $T_i$ has at most $(1-\alpha)n$ vertices, for some positive $\alpha$, then the same result holds under the much weaker assumptions that $p \geq (\log n)^2/(cn)$ and $\Delta(T_i) \leq c np / \log n$ for some~$c$ that depends only on $\alpha$ and $\varepsilon$. Our assumptions on maximum degrees of the trees are significantly weaker than those in all previously known approximate packing results.