AI 中文总结
研究了在渗流顶点转置图中，通过概率p保留边后，保证存在完美匹配和2-因子的条件及概率。
AI 中文摘要
设G是一个有n个顶点的连通简单顶点转置图，其度数为d，令G_p是通过以概率p保留每条边得到的随机生成子图。受Bedert、Draganić、Müyesser和Pavez-Signé关于渗流Cayley图中哈密顿回路的猜想的启发，我们为更广泛的连通顶点转置宿主图建立了其匹配和2-因子的结论。首先，对于每一个A>0，存在C=C(A)>0，使得对于每一个N≥n，条件(1-p)^d≤N^{-C}成立时，以概率至少为1-N^{-A}，当n为偶数时，G_p具有完美匹配；当n为奇数时，对于每个顶点v，G_p-v具有完美匹配。其次，若ν_2(H)是H中最大边不相交的生成2-因子数量，则对于每一个A>0和0<ε<1，若ε^2pd≥64(A+6)log(2n)，则有概率至少为1-n^{-A}，ν_2(G_p)≥⌊(1-ε)pd/2⌋。因此，pd≥Clogn保证了在所有连通顶点转置图上，既具有适当的匹配性质，又具有高概率的生成2-因子。如果pd/logn→∞，则ν_2(G_p)=(1+o(1))pd/2以高概率成立。系数1/2是最优的，因为每个生成2-因子包含n条边，而G_p包含约pnd/2条边。
英文摘要
Let $G$ be a connected simple vertex-transitive graph on $n$ vertices with degree $d$, and let $G_p$ be the random spanning subgraph obtained by retaining each edge of $G$ independently with probability $p$. Put $q:=1-p$. Motivated by a conjecture of Bedert, Draganić, Müyesser, and Pavez-Signé on Hamilton cycles in percolated Cayley graphs, we establish the corresponding matching and $2$-factor statements uniformly over the larger class of all connected vertex-transitive host graphs. For every $A>0$, if $q^d\le n^{-(5A+250)},$ then, with probability at least $1-n^{-A}$, the graph $G_p$ has a perfect matching when $n$ is even and is factor-critical when $n$ is odd. Separately, if $0<ε<1$ and $ ε^2pd\ge64(A+6)\log(2n), $ then, with probability at least $1-n^{-A}$, the graph $G_p$ contains at least \[ \left\lfloor\frac{(1-ε)pd}{2}\right\rfloor \] pairwise edge-disjoint spanning $2$-factors. Moreover, if $pd/\log n\to\infty$, then \[ ν_2(G_p)=(1+o(1))\frac{pd}{2} \] with high probability, which is asymptotically optimal, where $ν_2(G)$ is the maximum number of pairwise edge-disjoint spanning 2-factors in $G$. Thus logarithmic-order percolation already forces these two factor-theoretic consequences of Hamiltonicity beyond the Cayley setting.
