Abstract
For any positive integer n, we define the prime sum graph \(G_n=(V,E)\) of order n with the vertex set \(V=\{1,2,\cdots , n\}\) and \(E=\{ij: i+j \text{ is } \text{ prime }\}\). Filz in 1982 posed a conjecture that \(G_{2n}\) is Hamiltonian for any \(n\ge 2\), i.e., the set of integers \(\{1,2,\cdots , 2n\}\) can be represented as a cyclic rearrangement so that the sum of any two adjacent integers is a prime number. With a fundamental result in graph theory and a recent breakthrough on the twin prime conjecture, we prove that Filz’s conjecture is true for infinitely many cases.
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Notes
Douglas B. West’s page (https://faculty.math.illinois.edu/\(\sim \)west/openp/primegraph.html) quickly addresses that “it is easy to build a Hamiltonian cycle when \(2n+1\) and \(2n+3\) are both prime, but it is not even known if \(G_{2n}\) is Hamiltonian for infinitely many n”.
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Acknowledgements
We would like to thank Miklós Simonovits for valuable suggestions.
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H.-B. Chen: The author is supported by MOST 105-2115-M-035-006-MY2 and MOST 107-2115-M-035-003-MY2, and the research is partly done while the author was a member of Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan.
H.-L. Fu Supported by MOST 106-2115-M-009-008.
J.-Y. Guo Supported by MOST 106-2115-M-003-007.
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Chen, HB., Fu, HL. & Guo, JY. Hamiltonicity in Prime Sum Graphs. Graphs and Combinatorics 37, 209–219 (2021). https://doi.org/10.1007/s00373-020-02241-1
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DOI: https://doi.org/10.1007/s00373-020-02241-1