Discussiones
Mathematicae Graph Theory 21(1) (2001) 137-143
DOI: https://doi.org/10.7151/dmgt.1138
A NOTE ON A NEW CONDITION IMPLYING PANCYCLISM
Evelyne Flandrin LRI, Bât. 490 |
Antoni Marczyk Faculty of Applied Mathematics AGH |
Abstract
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 Kn/2,n/2.
Keywords: hamiltonian graphs, pancyclic graphs, cycles.
2000 Mathematics Subject Classification: 05C38, 05C45.
References
| [1] | J.A. Bondy, Pancyclic graphs I, J. Combin. Theory 11 (1971) 80-84, doi: 10.1016/0095-8956(71)90016-5. |
| [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. |
| [3] | J.A. Bondy and U.S.A. Murty, Graph Theory with Applications (Elsevier, North Holland, New York, 1976). |
| [4] | V. Chvátal, On Hamilton's ideals, J. Combin. Theory 12 (B) (1972) 163-168. |
| [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. |
| [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. |
| [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. |
| [8] | O. Ore, Note on hamilton circuits, Amer. Math. Monthly 67 (1960) 55, doi: 10.2307/2308928. |
| [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. |
| [10] | Z. Skupień, private communication. |
| [11] | R. Zhu, Circumference in 2-connected graphs, Qu-Fu Shiyuan Xuebao 4 (1983) 8-9. |
| [12] | L. Zhenhong, G. Jin and C. Wang, Two sufficient conditions for pancyclic graphs, Ars Combinatoria 35 (1993) 281-290. |
Received 15 November 2000
Revised 13 December 2000
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