Abstract
We prove that if G and H are graphs containing at least one edge each, then their lexicographic product G[H] is weakly pancyclic, i. e., it contains a cycle of every length between the length of a shortest cycle and that of a longest one. This supports some conjectures on locally connected graphs and on product graphs. We obtain an analogous result on even cycles in products G[H] that are bipartite. We also investigate toughness conditions on G implying that G[H] is hamiltonian (and hence pancyclic).
This is a preview of subscription content, access via your institution.
References
- 1.
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications (MacMillan, 1976)
- 2.
Diestel, R.: Graph Theory, Graduate Texts in Mathematics 173 (Springer, 1997)
- 3.
Ellingham, M.N., Zha, X.: Toughness, trees and walks. J. Graph Theory 33(3), 125–137 (2000)
- 4.
Jackson, B., Wormald, N.C.: k-walks of graphs. Australas. J. Combin. 2, 135–146 (1990)
- 5.
Kriesell, M.: A note on Hamiltonian cycles in lexicographical products. J. Autom. Lang. Comb. 2(2), 135–138 (1997)
- 6.
Ryjáček, Z.: Weak pancyclicity of locally connected graphs, Problem 416. Discrete Math. 272, 305–306 (2003)
Author information
Affiliations
Corresponding author
Additional information
Supported by the project LN00A056 of the Czech Ministry of Education
Rights and permissions
About this article
Cite this article
Kaiser, T., Kriesell, M. On the Pancyclicity of Lexicographic Products. Graphs and Combinatorics 22, 51–58 (2006). https://doi.org/10.1007/s00373-005-0639-7
Received:
Accepted:
Issue Date:
Keywords
- Lexicographic product
- Circumference
- Pancyclicity
- k-walk
- Toughness