# On the stability for pancyclicity

• Volume: 21, Issue: 2, page 223-228
• ISSN: 2083-5892

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## Abstract

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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 ${d}_{G}\left(u\right)+{d}_{G}\left(v\right). 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).

## How to cite

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Ingo Schiermeyer. "On the stability for pancyclicity." Discussiones Mathematicae Graph Theory 21.2 (2001): 223-228. <http://eudml.org/doc/270367>.

@article{IngoSchiermeyer2001,
abstract = {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 $d_G(u) + d_G(v) < k$. 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).},
author = {Ingo Schiermeyer},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {pancyclic graphs; stability; stable property},
language = {eng},
number = {2},
pages = {223-228},
title = {On the stability for pancyclicity},
url = {http://eudml.org/doc/270367},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Ingo Schiermeyer
TI - On the stability for pancyclicity
JO - Discussiones Mathematicae Graph Theory
PY - 2001
VL - 21
IS - 2
SP - 223
EP - 228
AB - 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 $d_G(u) + d_G(v) < k$. 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).
LA - eng
KW - pancyclic graphs; stability; stable property
UR - http://eudml.org/doc/270367
ER -

## References

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1. [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. Zbl0291.05109
2. [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. Zbl0331.05138
3. [3] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan Press, 1976). Zbl1226.05083
4. [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. Zbl0879.05042
5. [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.

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