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arXiv · 2607.20157

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

Mengyu Cao·Mei Lu·Xiamiao Zhao
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Abstract

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, Dragani\'c, M\"uyesser, and Pavez-Sign\'e 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 $0<\epsilon<1$ and $ \epsilon^2pd\ge64(A+6)\log(2n), $ then, with probability at least $1-n^{-A}$, the graph $G_p$ contains at least \[ \left\lfloor\frac{(1-\epsilon)pd}{2}\right\rfloor \] pairwise edge-disjoint spanning $2$-factors. Moreover, if $pd/\log n\to\infty$, then \[ \nu_2(G_p)=(1+o(1))\frac{pd}{2} \] with high probability, which is asymptotically optimal, where $\nu_2(G)$ 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.

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Mengyu Cao, Mei Lu, Xiamiao Zhao. 2026-07-22. Matchings and Near-Optimal 2-Factor Packings in Percolated Vertex-Transitive Graphs. https://arxiv.org/abs/2607.20157

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