A Stability Result of the Pósa Lemma

.For an integer \(α\) and a graph \(G\), the \(α\)-disintegration of \(G\) is the graph obtained from \(G\) by recursively deleting vertices of degree at most \(α\) until the resulting graph has no such vertex. Pósa proved that if a 2-connected graph contains a path on \(m≥ k\) vertices with end-vertices in its \( (k-1)/2\)-disintegration, then \(G\) contains a cycle of length at least \(k\). We prove that if a 2-connected graph contains a path on \(m≥ k\) vertices with end-vertices in its \( (k-3)/2\)-disintegration, then \(G\) contains either a cycle of length at least \(k\) or a specific family of graphs. As an application, we strengthen the Erdős–Gallai stablity theorem of Füredi, Kostochka, Luo, and Verstraëte.