We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u)+d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u)+d(v) ≥ n+z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m for each 3 ≤ m ≤ max (dC(u,v)+1, [(n+19)/13]), being the distance of u and v on a hamiltonian cycle of G.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1021,
author = {Ralph Faudree and Odile Favaron and Evelyne Flandrin and Hao Li},
title = {Pancyclism and small cycles in graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {16},
year = {1996},
pages = {27-40},
zbl = {0879.05042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1021}
}
Ralph Faudree; Odile Favaron; Evelyne Flandrin; Hao Li. Pancyclism and small cycles in graphs. Discussiones Mathematicae Graph Theory, Tome 16 (1996) pp. 27-40. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1021/
[000] [1] D. Amar, E. Flandrin, I. Fournier and A. Germa, Pancyclism in hamiltonian graphs, Discrete Math. 89 (1991) 111-131, doi: 10.1016/0012-365X(91)90361-5. | Zbl 0727.05039
[001] [2] A. Benhocine and A. P. Wojda, The Geng-Hua Fan conditions for pancyclic or hamilton-connected graphs, J. Combin. Theory (B) 42 (1987) 167-180, doi: 10.1016/0095-8956(87)90038-4. | Zbl 0613.05038
[002] [3] J.A. Bondy, Pancyclic graphs. I., J. Combin. Theory 11 (1971) 80-84, doi: 10.1016/0095-8956(71)90016-5. | Zbl 0183.52301
[003] [4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan Press, 1976). | Zbl 1226.05083
[004] [5] R. Faudree, O. Favaron, E. Flandrin and H. Li, The complete closure of a graph, J. Graph Theory 17 (1993) 481-494, doi: 10.1002/jgt.3190170406. | Zbl 0782.05046
[005] [6] E.F. Schmeichel and S.L. Hakimi, Pancyclic graphs and a conjecture of Bondy and Chvátal, J. Combin. Theory (B) 17 (1974) 22-34, doi: 10.1016/0095-8956(74)90043-4. | Zbl 0268.05120
[006] [7] 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
[007] [8] R.H. Shi, The Ore-type conditions on pancyclism of hamiltonian graphs, personal communication. | Zbl 0729.05031