On the stability for pancyclicity

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(d 6n 5 e − 5, n + t) ≤ s(P ) ≤ max(d 4n 3 e − 2, n + t), where t = 2dn+1 2 e − (n + 1).