Abstract
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.
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References
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)
Bondy, J.A.: Pancyclic graphs I. J. Combin. Theory Ser. B 11, 80–84 (1971)
Chen, G., Gould, R.J., Gu, X., Saito, A.: Cycles with a chord in dense graphs. Discret. Math. 341, 2131–2141 (2018)
Cream, M., Gould, R.J., Hirohata, K.: A note on extending Bondy’s meta-conjecture. Australas. J. Combin. 67(3), 463–469 (2017)
Faudree, R.J., Gould, R.J., Jacobson, M.S.: Pancyclic graphs and linear forests. Discret. Math. 309, 1178–1189 (2009)
Faudree, R.J., Gould, R.J., Jacobson, M.S., Lesniak, L.: Generalizing pancyclic and \(k\)-ordered graphs. Graphs Combin. 20, 291–309 (2004)
Finkel, D.: On the number of independent chorded cycles in a graph. Discret. Math. 308, 5265–5268 (2008)
Gould, R.J.: Graph Theory. Dover Publications Inc., Mineola (2012)
Gould, R.J., Hirohata, K., Horn, P.: On independent doubly chorded cycles. Discret. Math. 338, 2051–2071 (2015)
Hendry, G.R.T.: Extending cycles in graphs. Discret. Math. 85, 59–72 (1990)
Ore, O.: Note on Hamilton circuits. Am. Math. Mon. 67, 55 (1960)
Randerath, B., Schiermeyer, I., Tewes, M., Volkmann, L.: Vertex pancyclic graphs. Discret. Appl. Math. 120, 219–237 (2002)
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Cream, M., Gould, R.J. & Hirohata, K. Extending Vertex and Edge Pancyclic Graphs. Graphs and Combinatorics 34, 1691–1711 (2018). https://doi.org/10.1007/s00373-018-1960-2
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DOI: https://doi.org/10.1007/s00373-018-1960-2