Abstract
In 1989, Zhu, Li and Deng introduced the definition of implicit degree of a vertex v in a graph G, denoted by id(v). In this paper, we prove that if G is a 2-connected graph of order n such that id(u)+id(v) ≥ n for each pair of nonadjacent vertices u and v in G, then G is pancyclic unless G is bipartite, or else n = 4r, r ≥ 2 and G is isomorphic to F 4r .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benhocine, A., Wojda, A.: The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graphs. J. Combin. Theory, Ser. B, 42, 167–180 (1987)
Bondy, J.: Pancyclic graphs. J. Combin. Theory, Ser. B, 11, 80–84 (1971)
Bondy, J., Murty, U.: Graph Theory with Applications, Macmillan Press, London, 1976
Chen, B., Zhang, S.: An implicit degree condition for long cycles in 2-connected graphs. Appl. Math. Lett., 19, 1148–1151 (2006)
Dirac, G.: Some theorems on abstract graphs. Proc. London Math. Soc., 2, 69–81 (1952)
Ore, O.: Note on hamilton circuits. Amer. Math. Monthly, 67, 55 (1960)
Schmeichel, E., Hakimi, S.: Pancyclic graphs and a conjecture of Bondy and Chavátal. J. Combin. Theory, Ser. B, 17, 23–34 (1974)
Schmeichel, E., Hakimi, S.: A cycle structure theorem for hamiltonian graphs. J. Combin. Theory, Ser. B, 45, 99–107 (1988)
Zhu, Y., Li, H., Deng, X.: Implicit-degrees and circumferences. Graphs Combin., 5, 283–290 (1989)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, H., Cai, J.Q. An implicit degree ore-condition for pancyclicity of graphs. Acta. Math. Sin.-English Ser. 29, 1773–1780 (2013). https://doi.org/10.1007/s10114-013-1641-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-013-1641-2