Star decompositions and independent sets in random regular graphs

Abstract

A $k$-star decomposition of a graph is a partition of its edges into $k$-stars (i.e., $k$ edges with a common vertex). The paper studies the following problem: for what values of $k>d/2$ does the random $d$-regular graph have a $k$-star decomposition (asymptotically almost surely, provided that the number of edges is divisible by $k$)? Delcourt, Greenhill, Isaev, Lidick\'y, and Postle proposed the following conjecture. It is easy to see that a $k$-star decomposition necessitates the existence of an independent set of density $1-d/(2k)$. So let $k^{\mathrm{ind}}_d$ be the largest $k$ for which the random $d$-regular graph a.a.s. contains an independent set of this density. Clearly, $k$-star decompositions cannot exist for $k>k^{\mathrm{ind}}_d$. The conjecture suggests that this is essentially the only restriction: there is a threshold $k^\star_d$ such that $k$-star decompositions exist if and only if $k \leq k^\star_d$, and it (basically) coincides with the other threshold, i.e., $k^\star_d \approx k^{\mathrm{ind}}_d$. We confirm this conjecture for sufficiently large $d$ by showing that a $k$-star decomposition exists if $d/2< k < k^{\mathrm{ind}}_d$. In fact, we prove the existence even if $k=k^{\mathrm{ind}}_d$ for degrees $d$ with asymptotic density $1$.