Advertisement

Graph Theory pp 91-100 | Cite as

Some of My Favourite Conjectures: Local Conditions Implying Global Cycle Properties

  • Ortrud R. OellermannEmail author
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

Graph Theory is believed to have begun with the famous Königsberg Bridge Problem. Figure 1 shows a map of Königsberg as it appeared in the eighteenth century. The town was spanned by seven bridges passing over the river Pregel and connecting the four land masses on which Königsberg was built. The townsfolk amused themselves by taking walks through Königsberg, attempting to cross each of the seven bridges exactly once. In 1736 Euler put an end to their speculation that such a walk did not exist by giving a rigorous argument which proved their conjecture. Motivated by this problem, a graph is called eulerian if it has a closed walk that traverses each edge exactly once. It is well known that a connected graph is eulerian if and only if the degree of each vertex is even. The eulerian problem for connected graphs is thus easily solved by checking whether the degree of each vertex is even. The vertex analogue of eulerian graphs are the Hamiltonian graphs. These are graphs that have a cycle that passes through each vertex exactly once and are named after Sir William Rowan Hamilton who devised the Icosian Game for two players. One of the problems in the game required the first player to select a path of five vertices on the dodecahedron. The second player had to extend this path to a cycle that contained all 20 points of the dodecahedron. If a graph has a cycle that contains all its vertices such a cycle is called a Hamiltonian cycle. In contrast to the eulerian problem, the Hamilton cycle problem, i.e., the problem of determining whether a given graph has a Hamiltonian cycle, has no known simple solution. As a result many sufficient conditions for hamiltonicity have been established. In this chapter we will describe problems and conjectures that have their roots in the Hamilton cycle problem.
Fig. 1

A map of the Königsberg bridges

Notes

Acknowledgements

The author would like to thank the anonymous referee for his/her useful comments. This chapter was supported by an NSERC grant CANADA, Grant number RGPIN-2016-05237.

References

  1. 1.
    D. Amar, I. Fournier, A. Germa, Pancyclism in Chvátal-Erdős graphs. Graphs Combin. 7, 101–112 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    A. Asratian, Some properties of graphs with local Ore condition. ARS Combin. 41, 97–106 (1995)MathSciNetzbMATHGoogle Scholar
  3. 3.
    J.A. Bondy, Pancyclic graphs I. J. Combin. Theory 11, 80–84 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    J.A. Bondy, Pancyclic graphs: recent results, infinite and finite sets, in Colloquium Mathematical Society János Bolyai, Keszthely, Hungary (1973), pp. 181–187Google Scholar
  5. 5.
    J.A. Bondy, A remark on two sufficient conditions for Hamilton cycles. Discrete Math. 22, 191–194 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A. Borchert, S. Nicol, O.R. Oellermann, Global cycle properties of locally isometric graphs. Discrete Appl. Math. 205, 16–26 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. Borchert, S. Nicol, O.R. Oellermann, Global cycle properties in graphs with large minimum clustering coefficient. Quaestiones Math. 39(8), 1047–1070 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    G. Chartrand, R.E. Pippert, Locally connected graphs. Časopis Pěst. Mat. 99, 158–163 (1974)MathSciNetzbMATHGoogle Scholar
  9. 9.
    G. Chen, A. Saito, S. Shan, The existence of a 2-factor in a graph satisfying the local Chvátal–Erdös Condition. SIAM J. Discrete Math. 27(4), 1788–1799 (2014)zbMATHCrossRefGoogle Scholar
  10. 10.
    V. Chvátal, P. Erdös, A note on Hamiltonian circuits. Discrete Math. 2, 111–113 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    L. Clark, Hamiltonian properties of connected locally connected graphs. Congr. Numer. 32, 199–204 (1981)MathSciNetzbMATHGoogle Scholar
  12. 12.
    J. de Wet, Doctoral thesis: Local properties of graphs, University of South Africa (2017)Google Scholar
  13. 13.
    J. de Wet, S.A. van Aardt, Traceability of locally hamiltonian and locally traceable graphs. Discrete Math. Theor. Comput. Sci. 17(3), 245–262 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    J. de Wet, S.A. van Aardt, M. Frick, Hamiltonicity of locally traceable and locally hamiltonian graphs. Discrete Appl. Math. 236, 137–152 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    G.A. Dirac, Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    P. Erdős, Some problems in graph theory, in Hypergraph Seminar, Ohio State Univ., Columbus, Ohio, 1972. Lecture Notes in Mathematics, vol. 411 (Springer, Berlin, 1974), pp. 187–190Google Scholar
  17. 17.
    A. Goldner, F. Harary, Note on a smallest non-hamiltonian maximal planar graph. Bull Malaysian Math. Soc. 6(1), 41–42 (1975)Google Scholar
  18. 18.
    V.S. Gordon, Y.L. Orlovich, C. Potts, V.A. Strusevich, Hamiltonian properties of locally connected graphs with bounded vertex degree. Discrete Appl. Math. 159, 1759–1774 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    A.S. Hasratian, N.K. Khachatrian, Some localization theorems on hamiltonian circuits. J. Combin. Theory Ser. B 49, 287–294 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    G.R.T. Hendry, A strengthening of Kikust’s theorem. J. Graph Theory 13, 257–260 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    G.R.T. Hendry, Extending cycles in graphs. Discrete Math. 85, 59–72 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    B. Jackson, O. Ordaz, Chvátal–Erdös conditions for paths and cycles in graphs and digraphs: a survey. Discrete Math. 84, 241–254 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    P. Keevash, B. Sudakov, Pancyclicity of Hamiltonian and highly connected graphs. J. Combin. Theory Ser. B 100, 456–467 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    P.B. Kikust, The existence of a hamiltonian cycle in a regular graph of degree 5 (Rusian with Latvian summary). Latvian Math. Yearbook 16, 33–38 (1975)MathSciNetGoogle Scholar
  25. 25.
    C. Lee, B. Sudakov, Hamiltonicity, independence number, and pancyclicity. Eur. J. Combin. 33, 449–457 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    M. Li, D.G. Corneil, E. Mendelsohn, Pancyclicity and NP-completeness in planar graphs. Discrete Appl. Math. 98(3), 219–225 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    D. Lou, The Chvátal-Erdős condition in triangle free graphs. Discrete Math. 152, 253–257 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    L. Lovász, V. Neumann-Lara, M. Plummer, Mengerian theorems for paths of bounded length. Periodica Math. Hungarica 9(4), 269–276 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    D.J. Oberly, D.P. Sumner, Every connected, locally connected nontrivial graph with no induced claw is hamiltonian. J. Graph Theory 3, 351–356 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    O. Ore, Note on hamilton circuits. Am. Math. Mon. 67, 55 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    C.M. Pareek, On the maximum degree of locally hamiltonian non-hamiltonian graphs. Utilitas Math. 23, 103–120 (1983)MathSciNetzbMATHGoogle Scholar
  32. 32.
    C.M. Pareek, Z. Skupień, On the smallest non-hamiltonian locally hamiltonian graph. J. Univ. Kuwait (Sci.) 10, 9–16 (1983)MathSciNetzbMATHGoogle Scholar
  33. 33.
    A. Saito, Chvátal-Erdös theorem: old theorem with new aspects, in Computational Geometry and Graph Theory. Lecture Notes in Computer Science, vol. 4535 (Springer, Berlin, 2008), pp. 191–200CrossRefGoogle Scholar
  34. 34.
    Z. Skupień, Locally hamiltonian graphs and Kuratowski’s theorem. Bull. Acad. Poln. Sci. Sér. Sci. Math. Astronom. Phys. 13, 615–619 (1965)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Z. Skupień, Locally hamiltonian and planar graphs. Fund. Math. 58, 193–200 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    S.A. van Aardt, A.P. Burger, M. Frick, C. Thomassen, J. de Wet, Hamilton cycles in sparse locally connected graphs (Submitted)Google Scholar
  37. 37.
    S.A. van Aardt, M. Frick, O.R. Oellermann, J. de Wet, Global cycle properties in locally connected, locally traceable and locally hamiltonian graphs. Discrete Appl. Math. 205, 171–179 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    S.A. van Aardt, M. Frick, J. Dunbar, O.R. Oellermann, J. de Wet, On Saito’s conjecture and the Oberley-Sumner conjectures. Graphs Combin. (2017). https://doi.org/10.1007/s00373-017-1820-5 MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    D.B. West, Research problems. Discrete Math. 272, 301–306 (2003)CrossRefGoogle Scholar
  40. 40.
    A.A. Zykov, Problem 30, in Theory of Graphs and Applications (Proc. Symp. Smolenice, M. Fiedler (editor)), Prague (1964), pp. 164–165Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of WinnipegWinnipegCanada

Personalised recommendations