Geodesic
Uniquely hamiltonian graphs for many sets of degrees
Discrete mathematics & theoretical computer science, Tome 26 (2024) no. 3.

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We give constructive proofs for the existence of uniquely hamiltonian graphs for various sets of degrees. We give constructions for all sets with minimum 2 (a trivial case added for completeness), all sets with minimum 3 that contain an even number (for sets without an even number it is known that no uniquely hamiltonian graphs exist), and all sets with minimum 4, except {4}, {4,5}, and {4,6}. For minimum degree 3 and 4, the constructions also give 3-connected graphs. We also introduce the concept of seeds, which makes the above results possible and might be useful in the study of Sheehan's conjecture. Furthermore, we prove that 3-connected uniquely hamiltonian 4-regular graphs exist if and only if 2-connected uniquely hamiltonian 4-regular graphs exist. 
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     author = {Brinkmann, Gunnar and De Pauw, Matthias},
     title = {Uniquely hamiltonian graphs for many sets of degrees},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
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     number = {3},
     year = {2024},
     doi = {10.46298/dmtcs.13129},
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     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.13129/}
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Brinkmann, Gunnar; De Pauw, Matthias. Uniquely hamiltonian graphs for many sets of degrees. Discrete mathematics & theoretical computer science, Tome 26 (2024) no. 3. doi : 10.46298/dmtcs.13129. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.13129/

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