# Pancyclism and small cycles in graphs

Ralph Faudree; Odile Favaron; Evelyne Flandrin; Hao Li

Discussiones Mathematicae Graph Theory (1996)

- Volume: 16, Issue: 1, page 27-40
- ISSN: 2083-5892

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topRalph Faudree, et al. "Pancyclism and small cycles in graphs." Discussiones Mathematicae Graph Theory 16.1 (1996): 27-40. <http://eudml.org/doc/270583>.

@article{RalphFaudree1996,

abstract = {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]), $d_C(u,v)$ being the distance of u and v on a hamiltonian cycle of G.},

author = {Ralph Faudree, Odile Favaron, Evelyne Flandrin, Hao Li},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {cycle; hamiltonian; pancyclic; hamiltonian path; distance; hamiltonian cycle},

language = {eng},

number = {1},

pages = {27-40},

title = {Pancyclism and small cycles in graphs},

url = {http://eudml.org/doc/270583},

volume = {16},

year = {1996},

}

TY - JOUR

AU - Ralph Faudree

AU - Odile Favaron

AU - Evelyne Flandrin

AU - Hao Li

TI - Pancyclism and small cycles in graphs

JO - Discussiones Mathematicae Graph Theory

PY - 1996

VL - 16

IS - 1

SP - 27

EP - 40

AB - 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]), $d_C(u,v)$ being the distance of u and v on a hamiltonian cycle of G.

LA - eng

KW - cycle; hamiltonian; pancyclic; hamiltonian path; distance; hamiltonian cycle

UR - http://eudml.org/doc/270583

ER -

## References

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- [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. Zbl0268.05120
- [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. Zbl0607.05050
- [8] R.H. Shi, The Ore-type conditions on pancyclism of hamiltonian graphs, personal communication. Zbl0729.05031

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