Robustness for expander graphs

TL;DR

This paper investigates the robustness of expander graphs under random edge sparsification, establishing optimal conditions for the presence of Hamiltonian cycles, perfect matchings, and triangle factors with high probability.

Contribution

It provides new probabilistic thresholds for the existence of complex structures in sparsified expander graphs, addressing open problems and introducing iterative absorption and coupling techniques.

Findings

Hamiltonian cycles appear with high probability under optimal sparsification conditions.

Triangle factors are present with high probability in certain sparsified expanders, matching asymptotic bounds.

New methods include iterative absorption and coupling for triangles in random subgraphs.

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 $G$.