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- W3154261796 abstract "Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ Hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput [European J. Combin., 97 (2021), 103395], and that it holds for 4-regular graphs of connectivity 2 with the constant $144 < c$, which we believe to be minimal among all Hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every nonnegative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} cdot 3^{k+3}$ Hamiltonian cycles. We prove that for every $d ge 3$ there is an infinite family of Hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of Hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components." @default.
- W3154261796 created "2021-04-26" @default.
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- W3154261796 date "2022-03-01" @default.
- W3154261796 modified "2023-09-26" @default.
- W3154261796 title "Regular Graphs with Few Longest Cycles" @default.
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- W3154261796 doi "https://doi.org/10.1137/21m1414048" @default.
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