Abstract
For a graph G, we denote by p(G) and c(G) the number of vertices of a longest path and a longest cycle in G, respectively. For a vertex v in G, id(v) denotes the implicit degree of v. In this paper, we obtain that if G is a 2-connected graph on n vertices such that the implicit degree sum of any three independent vertices is at least n + 1, then either G contains a hamiltonian path, or c(G) ≥ p(G) − 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bondy, J.A. Longest paths and cycles in graphs of high degree. Research Report CORR 80-16, Univ. of Waterloo, 1980
Bondy, J.A., Murty, M.S.R. Graph Theory with Applications. Macmillan, London, 1976
Enomoto, H., Heuvel, J.V.D., Kaneko, A., Saito, A. Relative length of long paths and cycles in graphs with large degree sums. J. Graph Theory, 20: 213–225 (1995)
Enomoto, H., Kaneko, A., Tuza, Zs. P3-factors and covering cycles in graphs of minimum degree n/3, Colloq. Math. Soc. Jáinos Bolyai, 52: 213–220 (1987)
Nash-Williams, C.st.J.A. Edge-disjoint hamiltonian circuits in graphs with vertices of large valency. Academic Press, London, 1971
Zhu, Y.J., Li, H., Deng, X.T. Implicit–degrees and circumferences. Graphs Combin., 5: 283–290 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author is supported by the National Natural Science Foundation of China (Grant No. 11501322), the Postdoctoral Science Foundation of China (Grant No. 2015M571999) and the Natural Science Foundation of Shandong Province (Grant No. ZR2014AP002).
Rights and permissions
About this article
Cite this article
Cai, Jq., Li, H. An implicit degree condition for relative length of long paths and cycles in graphs. Acta Math. Appl. Sin. Engl. Ser. 32, 365–372 (2016). https://doi.org/10.1007/s10255-016-0561-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-016-0561-1