Packing trees of unbounded degrees in random graphs

TL;DR

This paper proves that large trees with unbounded degrees can be packed edge-disjointly into random graphs under weaker degree constraints than previously known, expanding the scope of tree packing in probabilistic graph models.

Contribution

It introduces new methods to pack large trees with unbounded degrees into $G_{n,p}$ under significantly weaker degree and probability conditions than prior work.

Findings

Edge-disjoint packing of large trees in $G_{n,p}$ with high probability

Weaker maximum degree conditions compared to previous results

Applicable to trees with a linear number of vertices less than $n$

Abstract

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.