Clique factors in pseudorandom graphs

TL;DR

This paper proves that certain pseudorandom graphs contain perfect clique packings, resolving a longstanding conjecture for triangles and extending to more general clique factors under specific pseudorandom conditions.

Contribution

It establishes new conditions under which pseudorandom graphs contain clique factors, including a proof of a conjecture for triangles and generalization to larger cliques.

Findings

Contains a $K_r$-factor under pseudorandom conditions

Resolves a conjecture for triangle factors from 2004

Guarantees the embedding of all graphs with max degree 2 under certain pseudorandomness

Abstract

An $n$-vertex graph is said to to be $(p,\beta)$-bijumbled if for any vertex sets $A,B\subseteq V(G)$, we have \[e(A,B)=p|A||B|\pm \beta \sqrt{|A||B|}.\] We prove that for any $3\leq r\in \mathbb{N}$ and $c>0$ there exists an $\varepsilon>0$ such that any $n$-vertex $(p,\beta)$-bijumbled graph with $n\in r \mathbb{N}$, $\delta(G)\geq cpn$ and $\beta \leq \varepsilon p^{r-1}n$, contains a $K_r$-factor. This implies a corresponding result for the stronger pseudorandom notion of $(n,d,\lambda)$-graphs. For the case of triangle factors, that is when $r=3$, this result resolves a conjecture of Krivelevich, Sudakov and Szab\'o from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition of $\beta=o(p^2n)$ actually guarantees that a $(p,\beta)$-bijumbled graph $G$ contains every graph on $n$ vertices with maximum degree at most 2.