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Counting Hamiltonian Cycles on Quartic 4-Vertex-Connected Planar Graphs

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Abstract

We show that counting Hamiltonian cycles on quartic 4-vertex-connected planar graphs is \(\#P\)-complete under many-one counting (“weakly parsimonious”) reductions, and that no Fully Polynomial-time Randomized Approximation Scheme (FPRAS) can exist for this integer counting problem unless \(NP = RP\).

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Acknowledgements

We thank the anonymous reviewers of our manuscript for their extraordinary detailed comments and suggestions, which have allowed us to significantly improve the presentation of our results. The author Robert D. Barish also wishes to acknowledge a Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS Research Fellow (18F18117) for support during the process of revising this manuscript.

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  1. Graduate School of Arts and Sciences, University of Tokyo, Meguro-ku Komaba 3-8-1, Tokyo, 153-8902, Japan

    Robert D. Barish & Akira Suyama

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  1. Robert D. Barish
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  2. Akira Suyama
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Correspondence to Robert D. Barish.

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Barish, R.D., Suyama, A. Counting Hamiltonian Cycles on Quartic 4-Vertex-Connected Planar Graphs. Graphs and Combinatorics 36, 387–400 (2020). https://doi.org/10.1007/s00373-019-02101-7

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  • DOI: https://doi.org/10.1007/s00373-019-02101-7

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