Graphs and Combinatorics

, Volume 34, Issue 6, pp 1691–1711 | Cite as

Extending Vertex and Edge Pancyclic Graphs

  • Megan CreamEmail author
  • Ronald J. Gould
  • Kazuhide Hirohata
Original Paper


A graph G of order \(n\ge 3\) is pancyclic if G contains a cycle of each possible length from 3 to n, and vertex pancyclic (edge pancyclic) if every vertex (edge) is contained on a cycle of each possible length from 3 to n. A chord of a cycle is an edge between two nonadjacent vertices of the cycle, and chorded cycle is a cycle containing at least one chord. We define a graph G of order \(n\ge 4\) to be chorded pancyclic if G contains a chorded cycle of each possible length from 4 to n. In this article, we consider extensions of the property of being chorded pancyclic to chorded vertex pancyclic and chorded edge pancyclic.


Chorded cycle Pancyclic Vertex pancyclic Edge pancyclic 


  1. 1.
    Bondy, J.A.: Pancyclic graphs. Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing, pp. 167–172. Louisiana State Univ., Baton Rouge (1971)Google Scholar
  2. 2.
    Bondy, J.A.: Pancyclic graphs I. J. Combin. Theory Ser. B 11, 80–84 (1971)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, G., Gould, R.J., Gu, X., Saito, A.: Cycles with a chord in dense graphs. Discret. Math. 341, 2131–2141 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cream, M., Gould, R.J., Hirohata, K.: A note on extending Bondy’s meta-conjecture. Australas. J. Combin. 67(3), 463–469 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Faudree, R.J., Gould, R.J., Jacobson, M.S.: Pancyclic graphs and linear forests. Discret. Math. 309, 1178–1189 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Faudree, R.J., Gould, R.J., Jacobson, M.S., Lesniak, L.: Generalizing pancyclic and \(k\)-ordered graphs. Graphs Combin. 20, 291–309 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Finkel, D.: On the number of independent chorded cycles in a graph. Discret. Math. 308, 5265–5268 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gould, R.J.: Graph Theory. Dover Publications Inc., Mineola (2012)zbMATHGoogle Scholar
  9. 9.
    Gould, R.J., Hirohata, K., Horn, P.: On independent doubly chorded cycles. Discret. Math. 338, 2051–2071 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hendry, G.R.T.: Extending cycles in graphs. Discret. Math. 85, 59–72 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ore, O.: Note on Hamilton circuits. Am. Math. Mon. 67, 55 (1960)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Randerath, B., Schiermeyer, I., Tewes, M., Volkmann, L.: Vertex pancyclic graphs. Discret. Appl. Math. 120, 219–237 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCedar Crest CollegeAllentownUSA
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  3. 3.Department of Electronic and Computer EngineeringNational Institute of Technology, Ibaraki CollegeIbarakiJapan

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