Abstract
A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, Posa 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 is not an element of E(G) contains a Hamiltonian cycle. Recently, Chau proved an Ore-type version of Posa's conjecture for graphs on n >= n(0) vertices using the regularity-blow-up method; consequently the n(0) 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 n(0), we believe that our method of proof will be of independent interest.