Clique-factors in sparse pseudorandom graphs

TL;DR

This paper proves that certain sparse pseudorandom regular graphs contain a perfect tiling of cliques of size t, extending understanding of graph decompositions in pseudorandom settings.

Contribution

It establishes conditions under which sparse pseudorandom regular graphs contain a K_t-factor, generalizing previous results to broader graph classes.

Findings

Existence of K_t-factors in certain sparse pseudorandom graphs

Conditions relating eigenvalues and degree for clique tilings

Extension of clique factor results to regular graphs with eigenvalue bounds

Abstract

We prove that for any $t\ge 3$ there exist constants $c>0$ and $n_0$ such that any $d$-regular $n$-vertex graph $G$ with $t\mid n\geq n_0$ and second largest eigenvalue in absolute value $\lambda$ satisfying $\lambda\le c d^{t}/n^{t-1}$ contains a $K_t$-factor, that is, vertex-disjoint copies of $K_t$ covering every vertex of $G$.