Hamiltonicity of graphs perturbed by a random geometric graph

TL;DR

This paper investigates the conditions under which the union of a deterministic graph with linear degrees and a random geometric graph becomes Hamiltonian, providing optimal bounds and a linear time algorithm for finding Hamilton cycles.

Contribution

It establishes an asymptotically optimal bound on the radius for Hamiltonicity in perturbed graphs and introduces a linear time algorithm for constructing Hamilton cycles.

Findings

Optimal bound on radius for Hamiltonicity

Hamilton cycle can be found in linear time

Union of deterministic and random geometric graphs is Hamiltonian under certain conditions

Abstract

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 $G^d(n,r)$, for any $d\geq1$. We obtain an asymptotically optimal bound on the minimum $r$ for which a.a.s. $H\cup G^d(n,r)$ is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.