Density Hajnal--Szemer\'{e}di theorem for cliques of size four

TL;DR

This paper advances the understanding of the density thresholds needed to guarantee multiple disjoint cliques of size four in large graphs, providing asymptotic classifications of extremal structures.

Contribution

It determines asymptotically five classes of extremal graphs for the case r=4 and proposes a candidate set for extremal structures for all r ≥ 5.

Findings

Identified five extremal graph classes for r=4.

Proposed a candidate set of extremal graphs for r ≥ 5.

Extended the density threshold problem to larger cliques.

Abstract

The celebrated Corr\'{a}di--Hajnal Theorem~\cite{CH63} and the Hajnal--Szemer\'{e}di Theorem~\cite{HS70} determined the exact minimum degree thresholds for a graph on $n$ vertices to contain $k$ vertex-disjoint copies of $K_r$, for $r=3$ and general $r \ge 4$, respectively. The edge density version of the Corr\'{a}di--Hajnal Theorem was established by Allen--B\"ottcher--Hladk\'y--Piguet~\cite{ABHP15} for large $n$. Remarkably, they determined the four classes of extremal constructions corresponding to different intervals of $k$. They further proposed the natural problem of establishing a density version of the Hajnal--Szemer\'{e}di Theorem: For $r \ge 4$, what is the edge density threshold that guarantees a graph on $n$ vertices contains $k$ vertex-disjoint copies of $K_r$ for $k \le n/r$. They also remarked, ``We are not even sure what the complete family of extremal graphs should be.'' We take the first step toward this problem by determining asymptotically the five classes of extremal constructions for $r=4$. Furthermore, we propose a candidate set comprising $r+1$ classes of extremal constructions for general $r \ge 5$.