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Graphs and Combinatorics

, Volume 34, Issue 2, pp 289–312 | Cite as

Minimal k-Connected Non-Hamiltonian Graphs

  • Guoli Ding
  • Emily Marshall
Original Paper

Abstract

In this paper, we explore minimal k-connected non-Hamiltonian graphs. Graphs are said to be minimal in the context of some containment relation; we focus on subgraphs, induced subgraphs, minors, and induced minors. When \(k=2\), we discuss all minimal 2-connected non-Hamiltonian graphs for each of these four relations. When \(k=3\), we conjecture a set of minimal non-Hamiltonian graphs for the minor relation and we prove one case of this conjecture. In particular, we prove all 3-connected planar triangulations which do not contain the Herschel graph as a minor are Hamiltonian.

Keywords

Hamilton cycles Graph minors 

References

  1. 1.
    Brinkmann, G., Larson, C., Souffriau, J., Van Cleemput, N.: Construction of planar \(4\)-connected triangulations. Ars Math. Contemp. 9, 145–149 (2015)MathSciNetMATHGoogle Scholar
  2. 2.
    Brousek, Jan: Minimal \(2\)-connected non-hamiltonian claw-free graphs. Discret. Math. 191, 57–64 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, Guantao, Xingxing, Yu., Zang, Wenan: The circumference of a graph with no \(K_{3, t}\)-minor, II. J. Comb. Theory Ser. B 102(6), 1211–1240 (2012)CrossRefMATHGoogle Scholar
  4. 4.
    Chvátal, V., Erdős, P.: A note on Hamiltonian circuits. Discret. Math. 2(2), 111–113 (1972)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Mark, E., Emily M., Kenta, O., Shoichi, T.: Hamiltonicity of planar graphs with a forbidden minor, Submitted for publication, (October 2016). arXiv:1610.06558
  6. 6.
    Jackson, B., Xingxing, Y.: Hamilton cycles in plane triangulations. J. Graph Theory 42(2), 138–150 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kotzig, A.: Regularly connected trivalent graphs without non-trivial cuts of cardinality 3, Acta Fac. Rerum Natur. Univ. Comenian. Math. 21, 1–14 (1968)MathSciNetMATHGoogle Scholar
  8. 8.
    Robertson, N., Seymour, P.: Graph minors IX: disjoint crossed paths. J. Comb. Theory Ser. B 49, 40–77 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Seymour, P.D.: Decomposition of regular matroids. J. Comb. Theory Ser. B 28, 305–359 (1980)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Tutte, W.T.: A theorem on planar graphs. Trans. Am. Math. Soc. 82, 99–116 (1956)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentLouisiana State UniversityBaton RougeUSA
  2. 2.Computer Science and Mathematics DepartmentArcadia UniversityGlensideUSA

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