Robustness for expander graphs

TL;DR

This work investigates robustness of spanning structures in sparse pseudorandom graphs by analyzing random sparsifications $G_p$ of $(n,d,\lambda)$-graphs. It proves that for $\lambda=o(d)$ and $d=\Omega(\log n)$, $G_p$ contains a Hamiltonian cycle (and a perfect matching when $n$ is even) with high probability once $p\ge(1+\gamma)\frac{\log n}{d}$, and it establishes a triangle-factor threshold under stronger density $d$ and spectral conditions using iterative absorption, spread distributions, and a new sparse triangle-coupling argument. The results address Frieze and Krivelevich’s robustness questions and extend them to sparse expander hosts, providing matching lower bounds and a framework that combines robust expansion, FKNP spread theory, and hypergraph coupling. The methods yield not only existence but also counting consequences for Hamilton cycles, perfect matchings, and triangle factors in pseudorandom graphs, highlighting the versatility of the absorption and spread-approach in sparse settings.

Abstract

We study robust versions of properties of $(n,d,λ)$-graphs, namely, the property of a random sparsification of an $(n,d,λ)$-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 $γ>0$, for sufficient large $n$, any $(n,d,λ)$-graph $G$ with $λ=o(d)$, $d=Ω(\log n)$ and $p\ge\frac{(1+γ)\log n}{d}$, $G\cap G(n,p)$ 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 $(n,d,λ)$-graph $G$ with $λ=o(\frac{d^2}{n})$, $d=Ω(n^{\frac{5}{6}}\log^{\frac{1}{2}}n)$ 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 $λ$ 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 $G$.

Theorems & Definitions (35)

• Theorem 1.1: draganic2024hamiltonicity
• Theorem 1.2: Friezerandomhamsparse
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: chernoff1952measure, Chernoff's bound
• Lemma 2.2: alon2016probabilistic, Expander mixing lemma
• Lemma 2.3: ferber2024hamiltonicity, mixing lemma for almost regular expanders
• Theorem 2.4: ferber2024hamiltonicity, random subgraphs of spectral expanders