Every 4-Connected Graph with Crossing Number 2 is Hamiltonian

Ozeki, Kenta; Zamfirescu, Carol T.

doi:10.1137/17m1138443

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2021

2022

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Cited by 2 publications

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“…Ozeki and the author [22] recently strengthened (iii) by showing that if a graph with crossing number 1 in which all vertex‐deleted subgraphs are hamiltonian contains at most one cubic vertex, then it is hamiltonian. It is unknown whether (iii) can be extended to crossing number 2.…”

Section: Introductionmentioning

confidence: 99%

On the hamiltonicity of a planar graph and its vertex‐deleted subgraphs

Zamfirescu

2022

Journal of Graph Theory

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Tutte proved that every planar 4-connected graph is hamiltonian. Thomassen showed that the same conclusion holds for the superclass of planar graphs with minimum degree at least 4 in which all vertex-deleted subgraphs are hamiltonian. We here prove that if in a planar n-vertex graph with minimum degree at least 4 at least n − 5 vertex-deleted subgraphs are hamiltonian, then the graph contains two hamiltonian cycles, but that for every c < 1 there exists a nonhamiltonian polyhedral n-vertex graph with minimum degree at least 4 containing cn hamiltonian vertex-deleted subgraphs. Furthermore, we study the hamiltonicity of planar triangulations and their vertex-deleted subgraphs as well as Bondy's meta-conjecture, and prove that a polyhedral graph with minimum degree at least 4 in which all vertex-deleted subgraphs are traceable, must itself be traceable.

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Section: Introductionmentioning

confidence: 99%