Hamiltonicity of graphs perturbed by a random geometric graph

We study Hamiltonicity in graphs obtained as the union of a deterministic n-vertex graph H with linear degrees and a d-dimensional random geometric graph Gd (n, r), for any d >= 1. We obtain an asymptotically optimal bound on the minimum r for which a.a.s. H boolean OR Gd (n, r) is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.