# 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 , 𝑟 ) , for any 𝑑 ≥ 1. Weobtain an asymptotically optimal bound on the minimum 𝑟 for which a.a.s. 𝐻 ∪ 𝐺 𝑑 ( 𝑛 , 𝑟 ) is Hamiltonian. Our proof provides a linear time algorithm to ﬁnd a Hamilton cycle in such graphs.
4 Citations
• Mathematics
• 2022
. Let 𝐺 be a graph obtained as the union of some 𝑛 -vertex graph 𝐻 𝑛 with minimum degree 𝛿 ( 𝐻 𝑛 ) ≥ 𝛼𝑛 and a 𝑑 -dimensional random geometric graph 𝐺 𝑑 ( 𝑛 , 𝑟 ) . We investigate under
• 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 Ω( n/α ( G )). We
• Mathematics
Random Structures &amp; Algorithms
• 2022
We obtain sufficient conditions for the emergence of spanning and almost‐spanning bounded‐degree rainbow trees in various host graphs, having their edges colored independently and uniformly at
• Mathematics, Computer Science
Random Structures &amp; Algorithms
• 2022
These proofs provide polynomial-time algorithms to ﬁnd cycles of any length and prove Hamiltonicity and pancyclicity in the graph obtained as the union of a determ-inistic 𝑛 -vertex graph with 𝛿 ( 𝐻 ) with a range of sublinear degrees.

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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 ) ≥ 𝛼𝑛 and a 𝑑 -dimensional random geometric graph 𝐺 𝑑 ( 𝑛 , 𝑟 ) . We investigate under
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These proofs provide polynomial-time algorithms to ﬁnd cycles of any length and prove Hamiltonicity and pancyclicity in the graph obtained as the union of a determ-inistic 𝑛 -vertex graph with 𝛿 ( 𝐻 ) with a range of sublinear degrees.
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We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ with linear minimum degree and the binomial
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Both results resolve a long standing conjecture, posed e.g. by Bollobas, that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2. • Mathematics, Computer Science Electron. J. Comb. • 2021 This note extends the result by Bohman, Frieze, and Martin on the threshold in$\mathbb{G}(n,p)$to sparser graphs with$\alpha=o(1)$, and discusses embeddings of bounded degree trees and other spanning structures in this model. • Mathematics Random Struct. Algorithms • 2021 This work closes the gap between the seminal work of Johansson, Kahn and Vu (which resolves the purely random case, i.e.,$\alpha =0$) and that of Hajnal and Szemeredi (which demonstrates that for$\alpha \geq 1-1/r$the initial graph already houses the desired perfect$K_r$-tiling). • Mathematics Random Struct. Algorithms • 2019 It is shown that with high probability the graph Gα∪G(n,C/n) contains copies of all spanning trees with maximum degree at most Δ simultaneously, where C depends only on α and Δ. • Mathematics SIAM J. Discret. Math. • 2021 It is shown that the effect of these random edges is significantly stronger, namely that one can almost surely find the$(2k + 1)\$-st power of a Hamilton cycle, which is the largest power one can guarantee in such a setting.