Triangle-factors in pseudorandom graphs

TL;DR

This paper proves that certain pseudorandom regular graphs with small second eigenvalue contain a triangle-factor, advancing understanding of the conditions under which such factors exist in graphs with near-optimal eigenvalue bounds.

Contribution

It establishes a near-optimal eigenvalue bound ensuring the existence of triangle-factors in pseudorandom regular graphs, extending previous constructions and bounds.

Findings

Graphs with small second eigenvalue contain triangle-factors.

The eigenvalue bound is close to the theoretical limit.

Provides conditions for 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 $\varepsilon d^2/(n \log n)$, for a small constant $\varepsilon > 0$, 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 Szab\'o, 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 $\lambda = \Omega(d^2 / n)$.