Ore-degree threshold for the square of a Hamiltonian cycle

A 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 $deg(u) + deg(v) \geq n$ for every $uv \notin E(G)$ contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type version of P\'osa's conjecture for graphs on $n\geq n_0$ 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.Comment: 24 pages, 1 figure. In addition to some fixed typos, this updated version contains a simplified "connecting lemma" in Section 3.

Local resilience for squares of almost spanning cycles in sparse random graphs

In 1962, P\'osa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $\delta(G)\ge 2n/3$. Only more than thirty years later Koml\'os, S\'ark\H{o}zy, and Szemer\'edi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $\epsilon > 0$ and $p=n^{-1/2+\epsilon}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+\epsilon)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2) one cannot hope for a resilience version for the square of a spanning cycle (as deleting all edges in the neighborhood of single vertex destroys this property) and (3) for $c<2/3$ a.a.s. $G_{n,p}$ contains a subgraph with minimum degree at least $cnp$ which does not contain the square of a path on $(1/3+c)n$ vertices