ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 21(1) (2001) 223-228
DOI: 10.7151/dmgt.1145


Ingo Schiermeyer

Fakultät für Mathematik und Informatik
Technische Universität Bergakademie Freiberg
D-09596 Freiberg, Germany


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 dG(u)+dG(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).

Keywords: pancyclic graphs, stability.

2000 Mathematics Subject Classification: 05C35, 05C38, 05C45.


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Received 15 November 2000
Revised 2 April 2001