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arXiv · 2402.06177

Hamiltonicity of Sparse Pseudorandom Graphs

Asaf Ferber·Jie Han·Dingjia Mao·Roman Vershynin
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Abstract

We show that every $(n,d,\lambda)$-graph contains a Hamilton cycle for sufficiently large $n$, assuming that $d\geq \log^{6}n$ and $\lambda\leq cd$, where $c=\frac{1}{70000}$. This significantly improves a recent result of Glock, Correia and Sudakov, who obtained a similar result for $d$ that grows polynomially with $n$. The proof is based on a new result regarding the second largest eigenvalue of the adjacency matrix of a subgraph induced by a random subset of vertices, combined with a recent result on connecting designated pairs of vertices by vertex-disjoint paths in $(n,d,\lambda)$-graphs. We believe that the former result is of independent interest and will have further applications.

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Asaf Ferber, Jie Han, Dingjia Mao, Roman Vershynin. 2024-02-09. Hamiltonicity of Sparse Pseudorandom Graphs. https://doi.org/10.1017/s0963548325000070

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