Filling the gap between Turán's theorem and Pósa's conjecture

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (such as Turán’s theorem) or on finding spanning subgraphs (such as Dirac’s theorem or more recently the Pósa conjecture). Only a few results give conditions to obtain some intermediate-sized subgraph. We contend that this neglect is unjustified. In this paper we investigate minimum-degree conditions under which a graph G contains squared paths and squared cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends results of Fan and Kierstead [J. Combin. Theory Ser. B 63 (1995), 55–64] and of Komlós, Sarközy, and Szemerédi [Random Structures Algorithms 9 (1996), 193–211] concerning containment of a spanning squared paths and a spanning squared cycle, respectively.