Hamiltonicity of Sparse Pseudorandom Graphs

We show that every (n,d,λ)-graph contains a Hamilton cycle for sufficiently large n, assuming that d≥ log¹⁰n and λ≤ cd, where c=1/9000. This significantly improves a recent result of Glock, Correia and Sudakov, who obtain a similar result for d that grows polynomially with n. The proof is based on the absorption technique combined with a new result regarding the second largest eigenvalue of the adjacency matrix of a subgraph induced by a random subset of vertices. We believe that the latter result is of an independent interest and will have further applications.