The asymptotic Sudoku number conjecture for random cubic graphs

Let $\mathcal{G}_{n,3}$ be the random $3$-regular graph on $n$ vertices, and let $s(G)$ denote the smallest size of a Sudoku set for a graph $G$. Here, a Sudoku set is a vertex subset admitting a partial $\chi(G)$-colouring with a unique extension to a $\chi(G)$-colouring of all of $G$. The notation a.a.s. means asymptotically almost surely as $n\to\infty$. Random cubic graph Sudoku-number conjecture. A.a.s.,

$$s(\mathcal{G}_{n,3})=(1+o(1))\frac{n}{4}.$$

The paper proves the upper bound $s(\mathcal{G}_{n,3})\leq(1+o(1))n/3$ a.a.s., while the deterministic lower bound is $s(\mathcal{G}_{n,3})\geq\lceil n/4+1/2\rceil$. The conjecture asserts that this lower bound is asymptotically sharp and therefore gives the smallest possible asymptotic Sudoku number for cubic graphs on $n$ vertices, but it remains open.