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For a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-universalif $G$ contains every graph in $\mathcal{F}$ as a (not necessarily induced)subgraph. For the family of all graphs on $n$ vertices and of maximum degree atmost two, $\mathcal{H}(n,2)$, we prove that there exists a constant $C$ suchthat for $p \geq C \left( \frac{\log n}{n^2} \right)^{\frac{1}{3}}$, thebinomial random graph $G(n,p)$ is typically $\mathcal{H}(n,2)$-universal. Thisbound is optimal up to the constant factor as illustrated in the seminal workof Johansson, Kahn, and Vu for triangle factors. Our result improvessignificantly on the previous best bound of $p \geq C \left(\frac{\logn}{n}\right)^{\frac{1}{2}}$ due to Kim and Lee. In fact, we prove the strongerresult that for the family of all graphs on $n$ vertices, of maximum degree atmost two and of girth at least $\ell$, $\mathcal{H}^{\ell}(n,2)$, $G(n,p)$ istypically $\mathcal H^{\ell}(n,2)$-universal when $p \geq C \left(\frac{\logn}{n^{\ell -1}}\right)^{\frac{1}{\ell}}$. This result is also optimal up to theconstant factor. Our results verify (in a weak form) a classical conjecture ofKahn and Kalai.

Optimal Threshold for a Random Graph to be 2-Universal