论文标题
大型诱导的树木在密集的随机图中
Large induced trees in dense random graphs
论文作者
论文摘要
Erdős和Palka在1983年开始对诱导树木的最大尺寸进行研究。他们证明，对于$ 0 <p <1 $ $ g_ {n，p} $的最大诱导树大小的每一个固定$ 0 <p <1 $，集中在$ 2 \ log_q（np）$左右，$ 2 \ log_q（np）$ abteryability，log_q（np）$ ablobiality，$ q =（$ q =（1-p =（1-p =（1-p）$）^$。 de la vega显示出$ p = c/n $的相同值的集中度，其中$ c $是一个很大的常数，他的证明也适用于所有较大的$ p $。我们表明，对于任何给定的树$ t $，具有有限的最高度和尺寸$ $（2-o（1））\ log_q（np）$，$ g_ {n，p} $包含$ t $的感应副本，具有$ n^{ - 1/2} \ ln^{10/9} n \ leq p \ leq p \ leq p \ leq leq leq p \ leq p \ leq p \ leq p \ leq p \ leq p \ leq 0.99 $ n^{ - 1/2} \ ln^{ - 1/1/1/1/20.99 $ 99。这是渐进的最佳选择。
Erdős and Palka initiated the study of the maximal size of induced trees in random graphs in 1983. They proved that for every fixed $0<p<1$ the size of a largest induced tree in $G_{n,p}$ is concentrated around $2\log_q (np)$ with high probability, where $q=(1-p)^{-1}$. De la Vega showed concentration around the same value for $p=C/n$ where $C$ is a large constant, and his proof also works for all larger $p$. We show that for any given tree $T$ with bounded maximum degree and of size $(2-o(1))\log_q(np)$, $G_{n,p}$ contains an induced copy of $T$ with high probability for $n^{-1/2}\ln^{10/9}n\leq p\leq 0.99$. This is asymptotically optimal.
