Bridged Hamiltonian Cycles in Sub-critical Random Geometric Graphs

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.