Star decompositions and independent sets in random regular graphs

TL;DR

This work analyzes when random $d$-regular graphs admit a decomposition of the edge set into edge-disjoint $k$-stars for $k>d/2$. It links $k$-star decompositions to independent sets of density $\alpha_{d,k}=1-\frac{d}{2k}$ and identifies a threshold near the independence-ratio limit $k^{\mathrm{ind}}_d$ as the critical barrier. The authors introduce thin independent sets and construct star decompositions from in-regular orientations of the remaining graph, establishing a.a.s. existence for $d/2<k\le k^{\mathrm{ind}}_d}$ for large $d$ (even at the endpoint under mild fractional-part conditions). A first-moment analysis alongside a generalized isoperimetric bound provides a robust framework to verify decompositions for specific degrees, supported by computational checks up to $d\le 3000$ with only a few exceptional cases. The results substantially advance understanding of star decompositions in random regular graphs and connect them tightly to the independent-set structure of these 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, Lidický, 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 $1$.

Theorems & Definitions (19)

• Conjecture 1.1
• Theorem 1.2
• Definition 2.1
• Proposition 2.2
• Definition 2.4
• Lemma 2.5
• proof
• Remark 2.6
• Lemma 4.1
• Definition 4.2