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Let n be sufficiently large and suppose that G is a digraph on n vertices where every vertex has in-and 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)

In this paper, we consider conditions that ensure a hamiltonian graph has a 2-factor with exactly k cycles. Brandt et al. proved that if G is a graph on n ≥ 4k vertices with minimum degree at least n 2 , then G contains a 2-factor with exactly k cycles; moreover this is best possible. Faudree et al. asked if there is some c < 1 2 such that δ(G) ≥ cn would… (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 G and H be balanced U, V-bigraphs on 2n vertices with ∆(H) ≤ 2. Let k be the number of components of H, δ U := min{deg G (u) : u ∈ U } and δ V := min{deg G (v) : v ∈ V }. We prove that if n is sufficiently large and δ U + δ V ≥ 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… (More)

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