Filling the gap between Turan's theorem and Posa's conjecture

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turán-type results) or on finding spanning subgraphs (Dirac-type results). In this paper, we are interested in finding intermediate-sized subgraphs. 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 the containment of a spanning squared path and a spanning squared cycle, respectively. Our results show that such minimum degree conditions constitute not merely an interpolation between the corresponding Turán-type and Dirac-type results, but exhibit other interesting phenomena.