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arXiv · 0906.3299

Filling the gap between Tur\'an's theorem and P\'osa's conjecture

Peter Allen·Julia Böttcher·Jan Hladký
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Abstract

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turan-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 Komlos, Sarkozy, and Szemeredi [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 Turan-type and Dirac-type results, but exhibit other interesting phenomena.

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Peter Allen, Julia Böttcher, Jan Hladký. 2009-06-18. Filling the gap between Tur\'an's theorem and P\'osa's conjecture. https://doi.org/10.1112/jlms%2Fjdr007

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