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Hamiltonicity of Graphs on Surfaces in Terms of Toughness and Scattering Number – A Survey

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Discrete and Computational Geometry, Graphs, and Games (JCDCGGG 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13034))

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Abstract

This paper aims to survey Hamiltonicity of graphs on surfaces, including stronger (e.g. Hamiltonian-connectedness) and weaker (e.g. containing Hamiltonian paths and spanning trees with certain conditions) properties. Toughness and scattering number conditions are necessary conditions for graphs to have such properties. Since every k-connected graph on a surface \(F^2\) satisfies some toughness and scattering number condition, we can expect that “every k-connected graph on a surface \(F^2\) satisfies the property \({ \mathcal {P}}\)”. We explain which triple \((k, F^2, { \mathcal {P}})\) makes the statement true from the viewpoint of toughness and scattering number of graphs.

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Notes

  1. 1.

    This statement is related to Malkevitch’s conjecture [66] stating that every 4-connected planar graph G contains a cycle of length l for any \(3 \le l \le |V(G)|\), possibly except for \(l = 4\). See also [21].

  2. 2.

    Their proof depends on the Four Color Theorem, which was still unsolved at that time. Without using the Four Color Theorem, the same result was proven by Goodey and Rosenfeld [48] with additional conditions and then finally, by Fleischner [40]. The most general result in this direction is the following due to Paulraja [82]; any 3-connected cubic graph is prism-Hamiltonian.

  3. 3.

    This is related to the map color theorem, see [84, 85].

  4. 4.

    Ding [31] gave a complete structure of \(K_{2,t}\)-minor-free graphs.

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Acknowledgments

I would thank the referees for carefully reading the paper and helpful comments. This work was supported by JSPS KAKENHI Grant Number JP18K03391.

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  1. Faculty of Environment and Information Sciences, Yokohama National University, Yokohama, Japan

    Kenta Ozeki

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  1. Kenta Ozeki
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  1. Tokyo University of Science, Tokyo, Japan

    Jin Akiyama

  2. Ateneo de Manila University, Quezon City, Philippines

    Reginaldo M. Marcelo

  3. Ateneo de Manila University, Quezon City, Philippines

    Mari-Jo P. Ruiz

  4. Osaka Prefecture University, Osaka, Japan

    Yushi Uno

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Ozeki, K. (2021). Hamiltonicity of Graphs on Surfaces in Terms of Toughness and Scattering Number – A Survey. In: Akiyama, J., Marcelo, R.M., Ruiz, MJ.P., Uno, Y. (eds) Discrete and Computational Geometry, Graphs, and Games. JCDCGGG 2018. Lecture Notes in Computer Science(), vol 13034. Springer, Cham. https://doi.org/10.1007/978-3-030-90048-9_7

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