Star decomposition threshold conjecture for random regular graphs

Let $d\geq 3$, let $\mathcal{G}_{N,d}$ be a uniformly random $d$-regular graph on $N$ vertices, and let $k_d^{\mathrm{ind}}=\lfloor\kappa_d(\alpha_d^\star)\rfloor$, where $\alpha_d^\star$ is the limiting independence ratio of $\mathcal{G}_{N,d}$. A $k$-star decomposition partitions the edges into stars with $k$ edges each. Star decomposition threshold conjecture. If

$$d/2<k\leq k_d^{\mathrm{ind}},$$

then $\mathcal{G}_{N,d}$ a.a.s. has a $k$-star decomposition as $N\to\infty$ whenever $Nd$ is divisible by $2k$, apart from $d=5$ and perhaps a small number of further exceptional degrees. The conjecture asserts that the independent-set obstruction is essentially the only restriction, connecting star decompositions with the extremal independent-set problem in random regular graphs; its status is unresolved in the supplied text.