Abstract
In this note, we show a sharp lower bound of \(\min \left\{{\sum\nolimits_{i = 1}^k {{d_G}({u_i}):{u_1}{u_2} \ldots {u_k}}} \right.\) is a path of (2-)connected G on its order such that (k-1)-iterated line graphs Lk−1(G) are hamiltonian.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A Benhocine, L Clark, N Köhler, H J Veldman. On circuits and pancyclic line graphs, J Graph Theory, 1986, 10(3): 411–425.
J A Bondy, U S R Murty. Graph Theory, New York: Springer, 2008.
R A Brualdi, R F Shanny. Hamiltonian line graphs, J Graph Theory, 1981, 5: 307–314.
P A Catlin, I T N Janakiraman, N Srinivasan. Hamilton cycles and closed trails in iterated line graphs, J Graph Theory, 1990, 14: 347–364.
G Chartrand. On hamiltonian line graphs, Trans Amer Math Soc, 1968, 134(4): 559–566.
L Clark. On hamiltonian line graphs, J Graph Theory, 1984, 8(2): 303–307.
G A Dirac. Some theorems on abstract graphs, Proc London Math Soc, 1952, 2: 69–81.
Z Ryjáček, G J Woeginger, L M Xiong. Hamiltonian index is NP-complete, Discrete Math, 2011, 159(4): 246–250.
M L Saražin. A simple upper bound for the hamiltonian index of a graph, Discrete Math, 1994, 134(1–3): 85–91.
H J Veldman. On dominating and spanning circuits in graphs, Discrete Math, 1994, 124(1–3): 229–239.
L M Xiong. The Hamiltonian index of a graph, Graphs Combin, 2001, 17(4): 775–784.
L M Xiong, H J Broersma, X L Li, M C Li. The Hamiltonian index of a graph and its branchbonds, Discrete Math, 2004, 285(1–3): 279–288.
L M Xiong, Z H Liu. Hamiltonian iterated line graphs, Discrete Math, 2002, 256(1–2): 407–422.
L M Xiong, Z Ryjacek, H J Broersma. On stability of the hamiltonian index under contractions and closures, J Graph Theory, 2005, 49: 104–115.
L M Xiong, Q Q Zhu. The hamiltonian index of a graph-A survey, J Jiangxi Normal University (Natural Science), 2014, 38: 229–235.(in Chinese)
Acknowledgement
We would like to express our sincere appreciation to reviewers and editors and thanks for their careful reading and valuable revision recommendations.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China(11871099). ©The Author(s) 2021.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articles Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articles Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Liu, Zm., Xiong, Lm. Degree sum conditions for hamiltonian index. Appl. Math. J. Chin. Univ. 36, 403–411 (2021). https://doi.org/10.1007/s11766-021-3885-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11766-021-3885-4