We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to .
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1138,
author = {Evelyne Flandrin and Hao Li and Antoni Marczyk and Mariusz Wo\'zniak},
title = {A note on a new condition implying pancyclism},
journal = {Discussiones Mathematicae Graph Theory},
volume = {21},
year = {2001},
pages = {137-143},
zbl = {0989.05064},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1138}
}
Evelyne Flandrin; Hao Li; Antoni Marczyk; Mariusz Woźniak. A note on a new condition implying pancyclism. Discussiones Mathematicae Graph Theory, Tome 21 (2001) pp. 137-143. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1138/
[000] [1] J.A. Bondy, Pancyclic graphs I, J. Combin. Theory 11 (1971) 80-84, doi: 10.1016/0095-8956(71)90016-5. | Zbl 0183.52301
[001] [2] J.A. Bondy and V. Chvátal, A method in graph theory, Discrete Math. 15 (1976) 111-136, doi: 10.1016/0012-365X(76)90078-9. | Zbl 0331.05138
[002] [3] J.A. Bondy and U.S.A. Murty, Graph Theory with Applications (Elsevier, North Holland, New York, 1976).
[003] [4] V. Chvátal, On Hamilton's ideals, J. Combin. Theory 12 (B) (1972) 163-168. | Zbl 0213.50803
[004] [5] G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952) 69-81, doi: 10.1112/plms/s3-2.1.69. | Zbl 0047.17001
[005] [6] G.H. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory (B) 37 (1984) 221-227, doi: 10.1016/0095-8956(84)90054-6. | Zbl 0551.05048
[006] [7] R. Faudree, O. Favaron, E. Flandrin and H. Li, Pancyclism and small cycles in graphs, Discuss. Math. Graph Theory 16 (1996) 27-40, doi: 10.7151/dmgt.1021. | Zbl 0879.05042
[007] [8] O. Ore, Note on hamilton circuits, Amer. Math. Monthly 67 (1960) 55, doi: 10.2307/2308928.
[008] [9] E.F. Schmeichel and S.L. Hakimi, A cycle structure theorem for hamiltonian graphs, J. Combin. Theory (B) 45 (1988) 99-107, doi: 10.1016/0095-8956(88)90058-5. | Zbl 0607.05050
[009] [10] Z. Skupień, private communication.
[010] [11] R. Zhu, Circumference in 2-connected graphs, Qu-Fu Shiyuan Xuebao 4 (1983) 8-9.
[011] [12] L. Zhenhong, G. Jin and C. Wang, Two sufficient conditions for pancyclic graphs, Ars Combinatoria 35 (1993) 281-290. | Zbl 0788.05059