Uniquely hamiltonian graphs for many sets of degrees
arxiv(2023)
摘要
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 and at least two
elements, where all degrees different from 4 are at least 10. 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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关键词
hamiltonian graphs,many sets
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