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arXiv 1607.07342math.CO
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随机图的无界度Packing树

Packing trees of unbounded degrees in random graphs

Asaf Ferber, Wojciech Samotij

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摘要：

在本文中，我们解决的问题，包装大型树木在$G_{n,p}$。特别地，我们证明了以下结果。设$T_1, \dotsc, T_N$是$n$ -顶点树，每个顶点树的最大度至多为$(np)^{1/6} / (\log n)^6$ .然后，可以以高概率找到随机图$G_{n,p}$中所有$T_i$的边不相交副本，条件是$p \geq (\log n)^{36}/n$和$N \le (1-\varepsilon)np/2$对于正常数$\varepsilon$。此外，如果每个$T_i$最多有$(1-α)n$顶点，对于某个正的$α$，则在$p \geq (\log n)^2/(cn)$和$Δ(T_i) \leq c np / \log n$的弱得多的假设下，对于某个仅依赖于$α$和$\varepsilon$的~ $c$，相同的结果也成立。我们对树的最大度的假设明显弱于所有已知的近似包装结果。

英文摘要：

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 $(1-α)n$ vertices, for some positive $α$, then the same result holds under the much weaker assumptions that $p \geq (\log n)^2/(cn)$ and $Δ(T_i) \leq c np / \log n$ for some~$c$ that depends only on $α$ and $\varepsilon$. Our assumptions on maximum degrees of the trees are significantly weaker than those in all previously known approximate packing results.

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