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**Every 4-connected graph with crossing number 2 is Hamiltonian.**
*(English)*
Zbl 1401.05175

Summary: A seminal theorem of Tutte states that 4-connected planar graphs are Hamiltonian. Applying a result of R. Thomas and X. Yu [J. Comb. Theory, Ser. B 62, No. 1, 114–132 (1994; Zbl 0802.05051)], one can show that every 4-connected graph with crossing number 1 is Hamiltonian. In this paper, we continue along this path and prove the titular statement. We also discuss the traceability and Hamiltonicity of 3-connected graphs with small crossing number and few 3-cuts, and present applications of our results.

### MSC:

05C45 | Eulerian and Hamiltonian graphs |

05C10 | Planar graphs; geometric and topological aspects of graph theory |

05C38 | Paths and cycles |

### Citations:

Zbl 0802.05051
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\textit{K. Ozeki} and \textit{C. T. Zamfirescu}, SIAM J. Discrete Math. 32, No. 4, 2783--2794 (2018; Zbl 1401.05175)

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### References:

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