A property P defined on all graphs of order n is said to be k-stable if for any graph of order n that does not satisfy P, the fact that uv is not an edge of G and that G + uv satisfies P implies . Every property is (2n-3)-stable and every k-stable property is (k+1)-stable. We denote by s(P) the smallest integer k such that P is k-stable and call it the stability of P. This number usually depends on n and is at most 2n-3. A graph of order n is said to be pancyclic if it contains cycles of all lengths from 3 to n. We show that the stability s(P) for the graph property “G is pancyclic” satisfies max(⎡6n/5]⎤-5, n+t) ≤ s(P) ≤ max(⎡4n/3]⎤-2,n+t), where t = 2⎡(n+1)/2]⎤-(n+1).
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1145,
author = {Ingo Schiermeyer},
title = {On the stability for pancyclicity},
journal = {Discussiones Mathematicae Graph Theory},
volume = {21},
year = {2001},
pages = {223-228},
zbl = {1009.05076},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1145}
}
Ingo Schiermeyer. On the stability for pancyclicity. Discussiones Mathematicae Graph Theory, Tome 21 (2001) pp. 223-228. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1145/
[000] [1] J.A. Bondy, Pancyclic graphs, in: R.C. Mullin, K.B. Reid, D.P. Roselle and R.S.D. Thomas, eds, Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium III (1971) 167-172. | Zbl 0291.05109
[001] [2] J.A. Bondy and V. Chvátal, A method in graph theory, Discrete Math. 15 (1976) 111-135, doi: 10.1016/0012-365X(76)90078-9. | Zbl 0331.05138
[002] [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan Press, 1976). | Zbl 1226.05083
[003] [4] 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
[004] [5] U. Schelten and I. Schiermeyer, Small cycles in Hamiltonian graphs, Discrete Applied Math. 79 (1997) 201-211, doi: 10.1016/S0166-218X(97)00043-7.