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Bridged Hamiltonian Cycles in Sub-critical Random Geometric Graphs

Author

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  • Ghurumuruhan Ganesan

    (HBNI
    Indian Institute of Science Education and Research (IISER))

Abstract

In this paper, we consider a random geometric graph (RGG) G on n nodes with adjacency distance rn 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 →∞, we can obtain a Hamiltonian cycle with maximum bridge length a constant multiple of rn 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.

Suggested Citation

  • Ghurumuruhan Ganesan, 2023. "Bridged Hamiltonian Cycles in Sub-critical Random Geometric Graphs," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 691-706, February.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:1:d:10.1007_s13171-021-00273-0
    DOI: 10.1007/s13171-021-00273-0
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