Hamiltonicity of regular sublinear expanders

We say that a $d$-regular graph is a $\gamma$-expander if for every not too large set of vertices $S$, there are at least $\gamma d |S|$ edges leaving $S$, and we say that a graph $G$ is $\gamma$-far from bipartite if at least $\gamma e(G)$ edges need to be removed to make it bipartite. We prove that there exists an absolute constant $K$ such that any $n$-vertex $d$-regular $\gamma$-expander with $d \ge (\gamma^{-1} \log n)^K$ is Hamiltonian, provided that it is bipartite or $\gamma$-far from bipartite. As applications, we obtain highly robust versions of recent important results on the Hamiltonicity of Cayley graphs and Kneser graphs. As part of our proof, we prove a random connecting lemma for sublinear expanders which might be of independent interest.