Triangle-factors in pseudorandom graphs

Abstract

We show that if the second eigenvalue $\lambda$ of a $d$-regular graph $G$ on $n\in 3\mathbb{Z}$ vertices is at most $\epsilon d^2/(n\log n)$, for a small constant $\epsilon >0$, then $G$ contains a triangle-factor. The bound on $\lambda$ is at most an $O(\log n)$ factor away from the best possible one: Krivelevich, Sudakov and Szabó, extending a construction of Alon, showed that for every function $d=d(n)$ such that $\Omega (n^{2/3})\le d\le n$ and infinitely many $n\in \mathbb{N}$ there exists a $d$-regular triangle-free graph $G$ with $\Theta (n)$ vertices and $\lambda =\Omega (d^2/n)$.

References in corpus (2)

• A sequence of triangle-free pseudorandom graphs
• An asymptotic multipartite Kühn-Osthus theorem

Cited by in corpus (4)

• Near-perfect clique-factors in sparse pseudorandom graphs
• Finding any given 2-factor in sparse pseudorandom graphs efficiently
• Hamiltonicity of Sparse Pseudorandom Graphs
• A Ramsey-Turán theory for tilings in graphs