Abstract. For an integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text] -disintegration of [Formula: see text] is the graph obtained from [Formula: see text] by recursively deleting vertices of degree at most [Formula: see text] until the resulting graph has no such vertex. Pósa proved that if a 2-connected graph contains a path on [Formula: see text] vertices with end-vertices in its [Formula: see text]-disintegration, then [Formula: see text] contains a cycle of length at least [Formula: see text]. We prove that if a 2-connected graph contains a path on [Formula: see text] vertices with end-vertices in its [Formula: see text]-disintegration, then [Formula: see text] contains either a cycle of length at least [Formula: see text] or a specific family of graphs. As an application, we strengthen the Erdős–Gallai stablity theorem of Füredi, Kostochka, Luo, and Verstraëte.

Abstract. The [Formula: see text]-blow-up of a given graph is obtained by replacing each edge by a clique of order [Formula: see text] where the new vertices of the cliques are distinct. Liu and Yuan determined the extremal graphs for the 3-blow-ups of a triangle and the [Formula: see text]-blow-ups of any complete graph with order at most [Formula: see text], respectively. We determine the Turán number for the [Formula: see text]-blow-ups of a complete graph with order at least [Formula: see text], completing the study of the extremal graphs for [Formula: see text]-blow-ups of complete graphs.