https://hal.inria.fr/hal-01218404Debiasio, LouisLouisDebiasioMU - Miami University [Ohio]Faizullah, SafiSafiFaizullahHewlett-Packard Department of Computer Science [Rutgers] - RU - Rutgers, The State University of New Jersey [New Brunswick] - Rutgers - Rutgers University SystemKhan, ImdadullahImdadullahKhanDepartment of Computer Science [Lahore]] - SSE - School of Science and Engineering [Lahore] - LUMS - Lahore University of Management SciencesOre-degree threshold for the square of a Hamiltonian cycleHAL CCSD2015Ore-degreeHamiltonian cyclecycle powers[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]Episciences Iam, Coordination2015-10-21 10:22:282021-10-19 16:05:552015-10-21 10:26:25enJournal articleshttps://hal.inria.fr/hal-01218404/document10.46298/dmtcs.2127application/pdf1A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n=2 contains a Hamiltonian cycle. In 1963, P´osa conjectured that every graph with minimum degree at least 2n=3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac’s theorem by proving that every graph with deg(u) + deg(v) ≥ n for every uv =2 E(G) contains a Hamiltonian cycle. Recently, Chˆau proved an Ore-type version of P´osa’s conjecture for graphs on n ≥ n0 vertices using the regularity–blow-up method; consequently the n0 is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller n0, we believe that our method of proof will be of independent interest.