Robust Hamiltonicity in families of Dirac graphs

A graph is called Dirac if its minimum degree is at least half of the number of vertices in it. Joos and Kim showed that every collection $\mathbb{G}=\{G_1,\ldots,G_n\}$ of Dirac graphs on the same vertex set $V$ of size $n$ contains a Hamilton cycle transversal, i.e., a Hamilton cycle $H$ on $V$ with a bijection $\phi:E(H)\rightarrow [n]$ such that $e\in G_{\phi(e)}$ for every $e\in E(H)$. In this paper, we determine up to a multiplicative constant, the threshold for the existence of a Hamilton cycle transversal in a collection of random subgraphs of Dirac graphs in various settings. Our proofs rely on constructing a spread measure on the set of Hamilton cycle transversals of a family of Dirac graphs. As a corollary, we obtain that every collection of $n$ Dirac graphs on $n$ vertices contains at least $(cn)^{2n}$ different Hamilton cycle transversals $(H,\phi)$ for some absolute constant $c>0$. This is optimal up to the constant $c$. Finally, we show that if $n$ is sufficiently large, then every such collection spans $n/2$ pairwise edge-disjoint Hamilton cycle transversals, and this is best possible. These statements generalize classical counting results of Hamilton cycles in a single Dirac graph.