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Pancyclic and Bipancyclic Graphs pp 1–7Cite as

Graphs

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Abstract

In this chapter we shall survey some basic ideas of graph theory. Most of our readers will already be familiar with these topics; those who are not, or wish to explore them further or to see proofs, should consult a recent book on the subject, such as Wallis (A beginner’s guide to graph theory, 2nd edn. Birkhäuser, Boston, MA (2007)) or West (An introduction to graph theory, 2nd edn. Prentice-Hall, Englewood Cliffs, NJ (2001)). Our main aim here is to ensure that we use all the terminology in the same way, and to standardize our notation.

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References

  1. D. Amar, E. Flandrin, I. Fournier, A. Germa, Pancyclism in Hamiltonian graphs. Discrete Math. 89, 111–131 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Bauer, E. Schmeichel, Hamiltonian degree conditions which imply a graph is pancyclic. J. Comb. Theory (B) 48, 111–116 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. J.A. Bondy, Pancyclic graphs I. J. Comb. Theory (B) 11, 80–84 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. J.A. Bondy, Pancyclic graphs: recent results. Colloq. Math. Soc. János Bolyai, 181–187 (1973)

    Google Scholar 

  5. J.A. Bondy, Longest paths and cycles in graphs of high degree. Research Report CORR 80-16, University of Waterloo, Waterloo, Ontario (1980)

    Google Scholar 

  6. R.L. Brooks, On coloring the nodes of a graph. Proc. Camb. Philos. Soc. 37, 194–197 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  7. W.K. Chen, On vector spaces associated with a graph. SIAM J. Appl. Math. 20, 385–389 (1971)

    Article  MathSciNet  Google Scholar 

  8. V. Chvátal, On Hamilton’s ideals. J. Comb. Theory (B) 12, 163–168 (1972)

    Article  MATH  Google Scholar 

  9. G.A. Dirac, Some theorems on abstract graphs. Proc. Lond. Math. Soc. 2, 69–81 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  10. R.C. Entringer, E.F. Schmeichel, Edge conditions and cycle structure in bipancyclic graphs. Ars. Comb. 26, 229–232 (1988)

    MathSciNet  MATH  Google Scholar 

  11. G.-H. Fan, New sufficient conditions for cycles in graphs. J. Comb. Theory (B) 37, 221–227 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. I. Fourneir, P. Fraisse, One a conjecture of Bondy. J. Comb. Theory (B) 39, 17–26 (1985)

    Article  MATH  Google Scholar 

  13. J.C. George, A. Marr, W.D. Wallis, Minimal pancyclic graphs. J. Comb. Math. Combin. Comput. 86, 125–133 (2013)

    MathSciNet  MATH  Google Scholar 

  14. W. Goddard, M.A. Henning, Note: pancyclicity of the prism. Discrete Math. 234, 139–142 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Gould, Graphs and vector spaces. J. Math. Phys. 37, 193–214 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Griffin, Minimal pancyclicity (to appear)

    Google Scholar 

  17. R. Häggkvist, Odd cycles of specified length in nonbipartite graphs. Ann. Discrete Math. 62, 89–99 (1982)

    MathSciNet  MATH  Google Scholar 

  18. W. Imrich, S. Klavzar, Product Graphs, Structure and Recognition (Wiley, New York, 2000)

    MATH  Google Scholar 

  19. S. Janson, T. Łuczak, A. Ruciński, Random Graphs (Wiley, New York, 2000)

    Book  MATH  Google Scholar 

  20. P.K. Jha, Kronicker products of paths and cycles: decomposition, factorization, and bi-pancyclicity. Discrete Math. 182, 153–167 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Khodkar, A. Peterson, C. Wahl, Z. Walsh, Uniquely bipancyclic graphs on more than 30 vertices. J. Comb. Math. Combin. Comput. (to appear)

    Google Scholar 

  22. C. Lai, Graphs without repeated cycle lengths. Aust. J. Comb. 27, 101–105 (2003)

    MathSciNet  MATH  Google Scholar 

  23. K. Markström, A note on uniquely pancyclic graphs. Aust. J. Comb. 44, 105–110 (2009)

    MathSciNet  MATH  Google Scholar 

  24. J. Mitchem, E.F. Schmeichel, Pancyclic and bipancyclic graphs – a survey, in Proceedings of the First Colorado Symposium on Graph Theory, ed. by F. Harary, J.S. Maybee (Wiley, New York, 1985), pp. 271–278.

    Google Scholar 

  25. O. Ore, Note on Hamilton circuits. Am. Math. Month. 67, 55 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  26. N.C.K. Phillips, W.D. Wallis, Uniquely bipancyclic graphs on thirty-two vertices. J. Discrete Math. Sci. Crypt. (to appear)

    Google Scholar 

  27. S. Ramachandran, R. Parvathy, Pancyclicity and extendability in strong products. J. Graph Theory 22, 75–82 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. E.F. Schmeichel, S.L. Hakimi, Pancyclic graphs and a conjecture of Bondy and Chvatal. J. Comb. Theory (B) 17, 22–34 (1974)

    Google Scholar 

  29. E.F. Schmeichel, J. Mitchem, Bipartite graphs with cycles of all even lengths. J. Graph Theory 6, 429–439 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y. Shi, Some theorems of uniquely pancyclic graphs. Discrete Math. 59, 167–180 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Y. Shi, The number of cycles in a Hamilton graph. Discrete Math. 133, 249–257 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  32. M.R. Sridharan, On an extremal problem concerning pancyclic graphs. J. Math. Phys. Sci. 12, 297–306 (1978)

    MathSciNet  MATH  Google Scholar 

  33. K. Thulasiraman, M.N.S. Swamy, Graphs: Theory and Algorithms (Wiley, New York, 1992)

    Book  MATH  Google Scholar 

  34. W.D. Wallis, A Beginner’s Guide to Graph Theory, 2nd edn. (Birkhäuser, Boston, MA, 2007)

    Book  MATH  Google Scholar 

  35. W.D. Wallis, Uniquely bipancyclic graphs. J. Comb. Math. Comb. Comput. (to appear)

    Google Scholar 

  36. D.B. West, An Introduction to Graph Theory, 2nd edn. (Prentice-Hall, Englewood Cliffs, NJ, 2001)

    Google Scholar 

  37. H. Whitney, On the abstract properties of linear dependence. Am. J. Math. 57, 509–533 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  38. C.T. Zamfirescu, (2)-Pancyclic graphs. Discrete Appl. Math. 161, 1128–1136 (2013)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Gordon State College, Barnesville, GA, USA

    John C. George

  2. Department of Mathematics, University of West Georgia, Carrollton, GA, USA

    Abdollah Khodkar

  3. Department of Mathematics, Southern Illinois University, Evansville, IN, USA

    W. D. Wallis

Authors
  1. John C. George
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  2. Abdollah Khodkar
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  3. W. D. Wallis
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George, J.C., Khodkar, A., Wallis, W.D. (2016). Graphs. In: Pancyclic and Bipancyclic Graphs. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-31951-3_1

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