Bridged Hamiltonian Cycles in Sub-critical Random Geometric Graphs

TL;DR

This paper investigates the construction of Hamiltonian cycles in sub-critical random geometric graphs by introducing and analyzing the use of bridges, which are additional edges outside the original graph, to connect cycles.

Contribution

It provides a probabilistic method to construct Hamiltonian cycles in near-threshold RGGs using bridges, with bounds on bridge length and quantity.

Findings

Hamiltonian cycles can be constructed with high probability using bridges.

Maximum bridge length is proportional to the adjacency distance r_n.

A small fraction of edges in the cycle are bridges.

Abstract

In this paper, we consider a random geometric graph (RGG)~\(G\) on~\(n\) nodes with adjacency distance~\(r_n\) just below the Hamiltonicity threshold and construct Hamiltonian cycles using additional edges called bridges. The bridges by definition do not belong to~\(G\) and we are interested in estimating the number of bridges and the maximum bridge length, needed for constructing a Hamiltonian cycle. In our main result, we show that with high probability, i.e. with probability converging to one as~\(n \rightarrow \infty,\) we can obtain a Hamiltonian cycle with maximum bridge length a constant multiple of~\(r_n\) and containing an arbitrarily small fraction of edges as bridges. We use a combination of backbone construction and iterative cycle merging to obtain the desired Hamiltonian cycle.