Near-perfect clique-factors in sparse pseudorandom graphs

TL;DR

This paper demonstrates that sparse pseudorandom graphs with certain eigenvalue bounds contain almost perfect clique-factors, advancing understanding of their structure and supporting existing conjectures.

Contribution

It proves the existence of near-complete clique-factors in sparse pseudorandom graphs under specific spectral conditions, extending prior results to larger cliques.

Findings

Graphs with eigenvalue bounds contain vertex-disjoint clique copies

Almost all vertices are covered by these clique-factors

Supports the conjecture on triangle-factors in sparse pseudorandom graphs

Abstract

We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$\lambda$ satisfying $\lambda\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint copies of $K_t$ covering all but at most $n^{1-1/(8t^4)}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Sz\'abo [\emph{Triangle factors in sparse pseudo-random graphs}, Combinatorica \textbf{24} (2004), pp.~403--426] that $(n,d,\lambda)$-graphs with $n\in 3\mathbb{N}$ and $\lambda\leq cd^{2}/n$ for a suitably small absolute constant~$c>0$ contain triangle-factors.