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arXiv · 2207.06891

Higher-dimensional counterexamples to Hamiltonicity

Bruno Benedetti·Marta Pavelka
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Abstract

For $d \ge 2$, we show that all graphs of $d$-polytopes have a Hamiltonian line graph if and only if $d \ne 3$: We exhibit a graph of a $3$-polytope on $252$ vertices whose line graph does not even have Hamiltonian paths. Adapting a construction by Gr\"unbaum and Motzkin, for large $n$ we also construct simple $3$-polytopes on $3n$ vertices in whose line graph any simple path is shorter than $10 n^{\alpha}$, for some constant $\alpha<1$. Moreover, we give four elementary counterexamples of plausible extensions to simplicial complexes of four famous results in Hamiltonian graph theory.

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Bruno Benedetti, Marta Pavelka. 2022-07-14. Higher-dimensional counterexamples to Hamiltonicity. https://doi.org/10.1007/s00373-025-02988-5

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