Louis DeBiasio

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Let K 3 4 − 2e denote the hypergraph consisting of two triples on four points. For an integer n, let t (n,K 3 4 − 2e) denote the smallest integer d so that every 3-uniform hypergraph G of order n with minimum pairdegree δ2(G) ≥ d contains n/4 vertex-disjoint copies of K 3 4 − 2e. Kühn and Osthus (J Combin Theory, Ser B 96(6) (2006), 767–821) proved that t(More)
Given a 3-graph H , let ex2(n,H) denote the maximum value of the minimum co-degree of a 3-graph on n vertices which does not contain a copy of H . Let F denote the Fano plane, which is the 3-graph {axx, ayy, azz , xyz , xyz, xyz, xyz }. Mubayi (2005) [14] proved that ex2(n, F) = (1/2 + o(1))n and conjectured that ex2(n, F) = ⌊n/2⌋ for sufficiently large n.(More)
In 1962 Pósa conjectured that every graph G on n vertices with minimum degree δ(G) ≥ 23n contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Pósa’s Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than 23n. Still in 1996, Komlós, Sárközy, and(More)
In 1960 Ghouila-Houri extended Dirac’s theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2, then D contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum(More)
Let n be sufficiently large and suppose that G is a digraph on n vertices where every vertex has inand outdegree at least n/2. We show that G contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is n/2 + 1. Our result is best possible and improves(More)
Let G and H be balanced U, V -bigraphs on 2n vertices with ∆(H) ≤ 2. Let k be the number of components of H, δU := min{degG(u) : u ∈ U} and δV := min{degG(v) : v ∈ V }. We prove that if n is sufficiently large and δU + δV ≥ n+k then G contains H. This answers a question of Amar in the case that n is large. We also show that G contains H even when δU + δV ≥(More)