Clique factors in random samplings of regular graphs

TL;DR

The paper proves that large regular graphs contain exponentially many subsets with a perfect clique factor, confirming a conjecture for sufficiently large graphs.

Contribution

It establishes a lower bound on the number of subsets forming a $K_r$-factor in large regular graphs, confirming a conjecture by Draganić, Keevash, and M"uyesser.

Findings

Existence of a constant c > 0 for large n

At least c2^{rn} subsets contain a K_r-factor

Confirms the conjecture for large regular graphs

Abstract

We show that for any integer $r\ge 2$, there exists a constant $c>0$ such that for every sufficiently large integer $n$, every $((r-1)n+1)$-regular graph $G$ on $rn$ vertices has at least $c2^{rn}$ subsets $S\subseteq V(G)$ such that $G[S]$ contains a $K_r$-factor. This confirms a conjecture of Dragani\'c, Keevash and M\"uyesser for large $n$ [Cyclic subsets in regular Dirac graphs. Int. Math. Res. Not., 2025(14): 1-16, 2025].