论文标题
加权Erdős-rényi图中的汉密尔顿周期
Hamilton cycles in weighted Erdős-Rényi graphs
论文作者
论文摘要
给定一个对称的$ n \ times n $矩阵$ p $带有$ 0 \ le p（u，v）\ le 1 $，我们通过独立（包括任何边缘$ \ {u {u {u，v \} $ in Pobitality $ p（u，v）$，将随机图定义$ g_ {n，p} $。对于$ k \ ge 1 $ ling $ \ mathcal {a} _k $是包含$ \ lfloor k/2 \ rfloor $ hamilton Cycles的属性，如果$ k $是奇怪的，则是一个完美的匹配，如果是奇怪的，则所有边缘diserdisjoint。在$ p $上的特征值条件以及其行总和上的条件，$ g_ {n，p} \ in \ mathcal {a} _k $在且仅当$ g_ {n，p} $具有最低$ k $ k $ whp时，就有很高的可能性。我们还提供了一个打击的时间版本。作为一种特殊情况，伪随机的$（n，d，μ）上的随机图形过程与某些常数$α> 0 $具有$μ\ le d（d/n）^α$的图形，一旦获得最小$ k $具有高概率的最小$ k $。
Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of containing $\lfloor k/2 \rfloor$ Hamilton cycles, and one perfect matching if $k$ is odd, all edge-disjoint. With an eigenvalue condition on $P$, and conditions on its row sums, $G_{n, P}\in \mathcal{A}_k$ happens with high probability if and only if $G_{n, P}$ has minimum degree $k$ whp. We also provide a hitting time version. As a special case, the random graph process on pseudorandom $(n, d, μ)$-graphs with $μ\le d(d/n)^α$ for some constant $α> 0$ has property $\mathcal{A}_k$ as soon as it acquires minimum degree $k$ with high probability.
