# 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 π -vertex graph π» withlinear degreesanda π -dimensionalrandom geometric graph πΊ π ( π , π ) , for any π β₯ 1. Weobtain an asymptotically optimal bound on the minimum π for which a.a.s. π» βͺ πΊ π ( π , π ) is Hamiltonian. Our proof provides a linear time algorithm to ο¬nd a Hamilton cycle in such graphs.Β

## 4 Citations

### Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph

- Mathematics
- 2022

. Let πΊ be a graph obtained as the union of some π -vertex graph π» π with minimum degree πΏ ( π» π ) β₯ πΌπ and a π -dimensional random geometric graph πΊ π ( π , π ) . We investigate underβ¦

### Minors, connectivity, and diameter in randomly perturbed sparse graphs

- Mathematics
- 2022

Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every n -vertex graph G contains a complete minor of order β¦( n/Ξ± ( G )). Weβ¦

### Rainbow trees in uniformly edgeβcolored graphs

- MathematicsRandom Structures & Algorithms
- 2022

We obtain sufficient conditions for the emergence of spanning and almostβspanning boundedβdegree rainbow trees in various host graphs, having their edges colored independently and uniformly atβ¦

### Hamiltonicity of graphs perturbed by a random regular graph

- Mathematics, Computer ScienceRandom Structures & Algorithms
- 2022

These proofs provide polynomial-time algorithms to ο¬nd cycles of any length and prove Hamiltonicity and pancyclicity in the graph obtained as the union of a determ-inistic π -vertex graph with πΏ ( π» ) with a range of sublinear degrees.

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### Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph

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. Let πΊ be a graph obtained as the union of some π -vertex graph π» π with minimum degree πΏ ( π» π ) β₯ πΌπ and a π -dimensional random geometric graph πΊ π ( π , π ) . We investigate underβ¦

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These proofs provide polynomial-time algorithms to ο¬nd cycles of any length and prove Hamiltonicity and pancyclicity in the graph obtained as the union of a determ-inistic π -vertex graph with πΏ ( π» ) with a range of sublinear degrees.

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