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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)
Erd˝ os, Gyárfás, and Pyber (1991) conjectured that every r-colored complete graph can be partitioned into at most r − 1 monochromatic components; this is a strengthening of a conjecture of Lovász (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a(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)