Robustness for expander graphs

Abstract

We study robust versions of properties of $(n,d,\lambda)$-graphs, namely, the property of a random sparsification of an $(n,d,\lambda)$-graph, where each edge is retained with probability $p$ independently. We prove such results for the containment problem of perfect matchings, Hamiltonian cycles, and triangle factors. These results address a series of problems posed by Frieze and Krivelevich. First we prove that given $\gamma>0$, for sufficient large $n$, any $(n,d,\lambda)$-graph $G$ with $\lambda=o(d)$, $d=\Omega(\log n)$ and $p\ge\frac{(1+\gamma)\log n}{d}$, $G\cap G(n,p)$ contains a Hamiltonian cycle (and thus a perfect matching if $n$ is even) with high probability. This result is asymptotically optimal. Moreover, we show that for sufficient large $n$, any $(n,d,\lambda)$-graph $G$ with $\lambda=o(\frac{d^2}{n})$, $d=\Omega(n^{\frac{5}{6}}\log^{\frac{1}{2}}n)$ and $p\gg d^{-1}n^{\frac{1}{3}}\log^{\frac{1}{3}} n$, $G\cap G(n,p)$ contains a triangle factor with high probability. Here, the restrictions on $p$ and $\lambda$ are asymptotically optimal. Our proof for the triangle factor problem uses the iterative absorption approach to build a spread measure on the triangle factors, and we also prove and use a coupling result for triangles in the random subgraph of an expander $G$ and the hyperedges in the random subgraph of the triangle-hypergraph of $G$.