Hamiltonicity of graphs perturbed by a random geometric graph

@inproceedings{Diaz2021HamiltonicityOG,
  title={Hamiltonicity of graphs perturbed by a random geometric graph},
  author={Alberto Espuny D'iaz},
  year={2021}
}
. We study Hamiltonicity in graphs obtained as the union of a deterministic 𝑛 -vertex graph 𝐻 withlinear degreesanda 𝑑 -dimensionalrandom geometric graph 𝐺 𝑑 ( 𝑛 , π‘Ÿ ) , for any 𝑑 β‰₯ 1. Weobtain an asymptotically optimal bound on the minimum π‘Ÿ for which a.a.s. 𝐻 βˆͺ 𝐺 𝑑 ( 𝑛 , π‘Ÿ ) is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.Β 

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. Let 𝐺 be a graph obtained as the union of some 𝑛 -vertex graph 𝐻 𝑛 with minimum degree 𝛿 ( 𝐻 𝑛 ) β‰₯ 𝛼𝑛 and a 𝑑 -dimensional random geometric graph 𝐺 𝑑 ( 𝑛 , π‘Ÿ ) . We investigate under…

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