Abstract
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 on an approximate result by Häggkvist and Thomason.
Highlights
A classical result on Hamilton cycles is Dirac’s theorem [3] which states that if G is a graph on n ≥ 3 vertices with minimum degree δ(G) ≥ n/2, G contains a Hamilton cycle
GhouilaHouri [4] proved an analogue of Dirac’s theorem for digraphs which guarantees that any digraph of minimum semidegree at least n/2 contains a consistently oriented Hamilton cycle (where the minimum semidegree δ0(G) of a digraph G is the minimum of all the in- and outdegrees of the vertices in G)
Haggkvist and Thomason [7] proved an approximate version of GhouilaHouri’s theorem for arbitrary orientations of Hamilton cycles. They showed that a minimum semidegree of n/2 + n5/6 ensures the existence of an arbitrary orientation of a Hamilton cycle in a digraph
Summary
A classical result on Hamilton cycles is Dirac’s theorem [3] which states that if G is a graph on n ≥ 3 vertices with minimum degree δ(G) ≥ n/2, G contains a Hamilton cycle. Haggkvist and Thomason [7] proved an approximate version of GhouilaHouri’s theorem for arbitrary orientations of Hamilton cycles. They showed that a minimum semidegree of n/2 + n5/6 ensures the existence of an arbitrary orientation of a Hamilton cycle in a digraph. This improved a result of Grant [5] for antidirected Hamilton cycles. [9] proved an approximate version of Theorem 1.2 for oriented graphs He showed that the semidegree threshold for an arbitrary orientation of a Hamilton cycle in an oriented graph is 3n/8 + o(n). Further related problems on digraph Hamilton cycles are discussed in [10]
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