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

Allen, Peter, Böttcher, Julia and Hladky, Jan (2011) Filling the gap between Turan's theorem and Posa's conjecture. Journal of the London Mathematical Society, 84 (2). pp. 269-302. ISSN 0024-6107

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Identification Number: 10.1112/jlms/jdr007


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.

Item Type: Article
Official URL:
Additional Information: © 2011 London Mathematical Society
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 28 May 2012 14:21
Last Modified: 20 Jun 2021 01:52

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