Near-perfect clique-factors in sparse pseudorandom graphs

Abstract

We prove that, for any t≥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 λ satisfying λ≤cdt−1/nt−2 contains (1−o(1))n/t vertex-disjoint copies of Kt. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), pp. 403–426] that (n,d,λ)-graphs with n∈3N and λ≤cd2 for a suitably small absolute constant c>0 contain triangle-factors.