Louis DeBiasio

Pósa's conjecture for graphs of order at least 2 × 10 8

In 1962 Posa conjectured that every graph G on n vertices with minimum degree \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb,amsfonts} \pagestyle{empty} \begin{document} \begin{align*}\delta(G)\ge \frac{2}{3}n\end{align*} \end{document} **image** contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Posas Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath,amssymb,amsfonts} \pagestyle{empty} \begin{document} \begin{align*}\frac{2}{3}n\end{align*} \end{document} **image** . Still in 1996, Komlos, Sarkozy, and Szemeredi proved Posas Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n ≥ n0, where n0 is a very large constant. Here we show without using these lemmas that n0:= 2 × 108 is sufficient. We are motivated by the recent work of Levitt, Sarkozy and Szemeredi, but our methods are based on techniques that were available in the 90s.

Tiling 3‐Uniform Hypergraphs With K43−2e

Let denote the hypergraph consisting of two triples on four points. For an integer n, let denote the smallest integer d so that every 3-uniform hypergraph G of order n with minimum pair-degree contains vertex-disjoint copies of . Kuhn and Osthus (J Combin Theory, Ser B 96(6) (2006), 767–821) proved that holds for large integers n. Here, we prove the exact counterpart, that for all sufficiently large integers n divisible by 4, A main ingredient in our proof is the recent “absorption technique” of Rodl, Rucinski, and Szemeredi (J. Combin. Theory Ser. A 116(3) (2009), 613–636).

Monochromatic cycle partitions of graphs with large minimum degree

Lehel conjectured that in every

An Extension of the Hajnal-Szemerédi Theorem to Directed Graphs

2

A Note on Bipartite Graph Tiling

-coloring of the edges of

2-Factors of Bipartite Graphs with Asymmetric Minimum Degrees

K_n

Improved degree conditions for 2-factors with k cycles in hamiltonian graphs

, there is a vertex disjoint red and blue cycle which span

Semi-degree threshold for anti-directed Hamiltonian cycles

V(K_n)

Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

. Łuczak, Rodl, and Szemeredi proved Lehels conjecture for large

On the co-degree threshold for the Fano plane

n