Matchings and Near-Optimal 2-Factor Packings in Percolated Vertex-Transitive Graphs

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 independently with probability $p$. Motivated by a conjecture of Bedert, Draganić, Müyesser, and Pavez-Signé on Hamilton cycles in percolated Cayley graphs, we establish its matching and $2$-factor consequences for the larger class of all connected vertex-transitive host graphs. First, for every $A>0$, there is $C=C(A)>0$ such that, for every $N\ge n$, the condition $(1-p)^d\le N^{-C}$ implies, with probability at least $1-N^{-A}$, that $G_p$ has a perfect matching when $n$ is even and that $G_p-v$ has a perfect matching for every vertex $v$ when $n$ is odd. Second, if $\nu_2(H)$ is the maximum number of pairwise edge-disjoint spanning $2$-factors in $H$, then, for every $A>0$ and $0<\epsilon<1$, \[
\epsilon^2pd\ge64(A+6)\log(2n) \] implies \[
\mathbb P\left(
\nu_2(G_p)\ge\left\lfloor(1-\epsilon)\frac{pd}{2}\right\rfloor
\right)\ge1-n^{-A}. \] Consequently, $pd\ge C\log n$ guarantees both the appropriate matching property and a spanning $2$-factor with high probability, uniformly over all connected vertex-transitive graphs. If $pd/\log n\to\infty$, then $\nu_2(G_p)=(1+o(1))pd/2$ with high probability. The coefficient $1/2$ is best possible because each spanning $2$-factor contains $n$ edges, whereas $G_p$ contains about $pnd/2$ edges.