Bedert–Drăganić–Müyesser–Pavez-Signé random Cayley-graph Hamilton-cycle conjecture

Let $G$ be a connected Cayley graph of order $n$ and degree $d$. For $p\in[0,1]$, let $G_p$ be the random spanning subgraph obtained by retaining each edge independently with probability $p$.

Bedert–Drăganić–Müyesser–Pavez-Signé conjecture. There is an absolute constant $C$ such that, for every connected Cayley graph $G$ of order $n$ and degree $d$, the condition

$$p\ge C\frac{\log n}{d}$$

implies that $G_p$ has a Hamilton cycle with high probability.

This is a random analogue of the Cayley-graph Hamilton-cycle conjecture. The paper presents it as an open problem and proves related matching and $2$-factor consequences for the larger class of connected vertex-transitive host graphs.