An exact minimum degree condition for Hamilton cycles in oriented graphs

Abstract

Abstract. We show that every sufficiently large oriented graph G with δcontains a Hamilton cycle. This is best possible and solves a problem of Thomassen from 1979.

A Dirac-Type Result on Hamilton Cycles in Oriented Graphs

“…Since this paper was written, we have used some of the tools and methods to obtain an exact version of Theorem 1 (but not of Theorems 3 and 4) for large oriented graphs [12] as well as an approximate analogue of Chvátal's theorem on Hamiltonian degree sequences for digraphs [17]. See [13] for related results about short cycles and pancyclicity for oriented graphs.…”

Finding Hamilton cycles in robustly expanding digraphs

“…Our parallel algorithmic version of Theorem 2 is best possible not only in the sense that there are oriented graphs G with δ 0 (G) = (3|G|−4)/8 −1 which are not Hamiltonian, (see [14] for examples) but also in the following sense. Given an oriented graph G on n vertices with δ 0 (G) ηn where 0 < η < 3/8, it is NPcomplete to decide whether G contains a Hamilton cycle.…”

“…Theorem 2 For every α > 0 there exists an integer n 0 = n 0 (α) such that for every oriented graph G of order n n 0 the following hold: Finally, the conjecture of Häggkvist was proved for all large enough oriented graphs by Keevash, Kühn and Osthus [14].…”

Directed Hamilton Cycles in Digraphs and Matching Alternating Hamilton Cycles in Bipartite Graphs

“…The readers could refer to the surveys of Gould ([17] and [18]), Kawarabayashi ([22]) and Broersma ([11]) to trace the development in this field. Recently, approximate solutions of many traditional Hamiltonian problems and conjectures in digraphs came forth ( [24], [23], [12] and [26]), which are surveyed by Kühn and Osthus ([25]). …”