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Clique factors in random samplings of regular graphs

Wanting Sun, Shunan Wei, Donglei Yang

TL;DR

This work resolves the robustness question for the Hajnal–Szemerédi K_r-factor by showing that for every r≥2 there exists c>0 such that, for all sufficiently large n, any ((r-1)n+1)-regular graph on rn vertices has at least c2^{rn} vertex subsets S for which G[S] contains a K_r-factor. The authors split the analysis into non-extremal and extremal cases, introducing a notion of good partitions and a balancing lemma to ensure the existence of a K_r-factor in the random induced subgraph G[S], with explicit probability bounds. A sequence of technical lemmas—constructing good partitions, balancing bad vertices, and a vertex-cover bound—underpin the extremal case, while existing results on multipartite Hajnal–Szemerédi tilings drive the core tiling arguments. The main quantitative outcome is a concrete constant c ≤ (40r^2)^{-r}, establishing a positive density of favorable vertex subsets and confirming the Draganić–Keevash–Müyesser conjecture for large n.

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 Draganić, Keevash and Müyesser for large $n$ [Cyclic subsets in regular Dirac graphs. Int. Math. Res. Not., 2025(14): 1-16, 2025].

Clique factors in random samplings of regular graphs

TL;DR

This work resolves the robustness question for the Hajnal–Szemerédi K_r-factor by showing that for every r≥2 there exists c>0 such that, for all sufficiently large n, any ((r-1)n+1)-regular graph on rn vertices has at least c2^{rn} vertex subsets S for which G[S] contains a K_r-factor. The authors split the analysis into non-extremal and extremal cases, introducing a notion of good partitions and a balancing lemma to ensure the existence of a K_r-factor in the random induced subgraph G[S], with explicit probability bounds. A sequence of technical lemmas—constructing good partitions, balancing bad vertices, and a vertex-cover bound—underpin the extremal case, while existing results on multipartite Hajnal–Szemerédi tilings drive the core tiling arguments. The main quantitative outcome is a concrete constant c ≤ (40r^2)^{-r}, establishing a positive density of favorable vertex subsets and confirming the Draganić–Keevash–Müyesser conjecture for large n.

Abstract

We show that for any integer , there exists a constant such that for every sufficiently large integer , every -regular graph on vertices has at least subsets such that contains a -factor. This confirms a conjecture of Draganić, Keevash and Müyesser for large [Cyclic subsets in regular Dirac graphs. Int. Math. Res. Not., 2025(14): 1-16, 2025].
Paper Structure (12 sections, 16 theorems, 68 equations)

This paper contains 12 sections, 16 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The Hajnal-Szemerédi theorem
  3. Notation
  4. Overview
  5. Non-extremal case
  6. Extremal case
  7. Proof of Theorem \ref{['lem:extremal-case']}
  8. Proof of Lemmas
  9. Proof of Lemma \ref{['lem:construct good-partition']}
  10. Proof of Lemma \ref{['lem:covering bad vertices for balance']}
  11. Proof of Lemma \ref{['vertex-cover']}
  12. Concluding remarks

Key Result

Theorem 1.2

Let $G$ be an $n$-vertex graph with $n\in r\mathbb{N}$. If $\delta(G)\ge \left(1-\frac{1}{r}\right)n$, then $G$ contains a $K_r$-factor.

Theorems & Definitions (39)

  • Conjecture 1.1: Erdos1999
  • Theorem 1.2: CorradiHajnalSz
  • Conjecture 1.3: Keevash2025
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: Non-extremal case
  • Theorem 2.2: Extremal case
  • Definition 2.3
  • Definition 2.4: Good partition
  • Theorem 3.1: gan2024
  • ...and 29 more