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Triangle-factors in pseudorandom graphs (1805.09710v1)
Published 24 May 2018 in math.CO
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 $\varepsilon d2/(n \log n)$, 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 $\lambda = \Omega(d2 / n)$.