Oriented Hamilton Cycles in Oriented Graphs

Abstract

We show that, for every ∈ > 0, an oriented graph of order n will contain n -cycles of every orientation provided each vertex has indegree and outdegree at least (5/12 + ∈) n and n > n 0 (∈) is sufficiently large. Introduction Dirac's theorem states that every graph G with minimum degree δ( G ) ≥ | G |/2 has a hamilton cycle. The simplest analogue for digraphs is given by the theorem of Ghouila-Houri. Given a digraph G of order n and a vertex v ∈ G , we denote the outdegree of v by d + ( v ) and the indegree by d − ( v ). We also define d ° ( v ) to be min{ d + ( v ), d − ( v )}, and δ ° ( G ) to be min{ d ° ( v ): v ∈ G }. Ghouila-Houri's theorem implies that G contains a directed hamilton cycle if d ° ( G )(G) ≥ n /2. Only recently has a constant c oriented graph satisfying δ ° ( G ) > cn has a directed hamilton cycle; Haggkvist has shown that c = (½ − 2 −15 ) will suffice. He also showed that the condition δ ° ( G ) > n /3 proposed by Thomassen is inadequate to guarantee a hamilton cycle, and conjectured that δ ° ( G ) ≥ 3 n /8 is sufficient. When considering hamilton cycles in digraphs there is no reason to stick to directed cycles only; we might ask for any orientation of an n -cycle. For tournaments G , Thomason has shown that G will contain every oriented cycle (except the directed cycle if G is not strong) regardless of the degrees, provided n is large.