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.

Monochromatic cycle power partitions

Improving our earlier result we show that for every integer k≥1 there exists a c(k) such that in every 2-colored complete graph apart from at most c(k) vertices the vertex set can be covered by 200k2logk vertex disjoint monochromatic kth powers of cycles. © 2016 Elsevier B.V

Embedding graphs having Ore-degree at most five

Let $H$ and $G$ be graphs on $n$ vertices, where $n$ is sufficiently large. We prove that if $H$ has Ore-degree at most 5 and $G$ has minimum degree at least $2n/3$ then $H\subset G.$Comment: accepted for publication at SIAM J. Disc. Mat

Minimum degrees for powers of paths and cycles

We study minimum degree conditions under which a graph $G$ contains $k^{th}$ power of paths and cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends a result of Allen, B\"ottcher and Hladk\'y concerning the containment of squares of paths and squares of cycles of arbitrary specified lengths and settles a conjecture of theirs in the affirmative.Comment: 69 pages, 3 figures. arXiv admin note: text overlap with arXiv:0906.3299 by other author

Stability for vertex cycle covers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac’s Theorem: for any integer k > 2, if G is an n-vertex graph with minimum degree at least n/k, then there are k − 1 cycles in G that together cover all the vertices. This is tight in the sense that there are n-vertex graphs that have minimum degree n/k − 1 and that do not contain k − 1 cycles with this property. A concrete example is given by In,k = Kn \ K(k−1)n/k+1 (an edge-maximal graph on n vertices with an independent set of size (k − 1)n/k + 1). This graph has minimum degree n/k − 1 and cannot be covered with fewer than k cycles. More generally, given positive integers k1, . . . , kr summing to k, the disjoint union Ik1n/k,k1 +· · ·+Ikrn/k,kr is an n-vertex graph with the same properties. In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph G has n vertices and minimum degree nearly n/k, then it either contains k − 1 cycles covering all vertices, or else it must be close (in ‘edit distance’) to a subgraph of Ik1n/k,k1 + · · · + Ikrn/k,kr , for some sequence k1, . . . , kr of positive integers that sum to k. Our proof uses Szemer´edi’s Regularity Lemma and the related machinery