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

TL;DR

This paper establishes precise minimum degree thresholds for large graphs to contain intermediate-sized squared paths and cycles, bridging the gap between Turán-type and Dirac-type extremal results.

Contribution

It extends previous results by determining exact degree conditions for the existence of squared paths and cycles of arbitrary lengths in large graphs.

Findings

Identifies exact degree thresholds for squared paths and cycles

Shows these thresholds interpolate between known extremal results

Reveals new phenomena beyond simple interpolation

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.