A Dirac-Type Result on Hamilton Cycles in Oriented Graphs

Abstract

Abstract. We show that for each α > 0 every sufficiently large oriented graph G with δ + (G), δ − (G) ≥ 3|G|/8 + α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen [21]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G) + δ + (G) + δ − (G) ≥ 3|G|/2 + α|G|. Up to the term α|G| this confirms a conjecture of Häggkvist [10]. We also prove an Ore-type theorem for oriented graphs.

An exact minimum degree condition for Hamilton cycles in oriented graphs

“…In Section 5 we prove a lemma that enables us to find a Hamilton cycle when G is not structurally similar to the extremal example. The argument in this case is based on that of [12]. Then we prove our main theorem in the final section.…”

“…The next lemma (which is essentially from [12]) shows that by discarding edges with appropriate probabilities one can go over to a reduced oriented graph R ⊆ R ′ which still inherits the minimum degree and density of G. …”

Finding Hamilton cycles in robustly expanding digraphs

“…The fact that the graphs in Theorem 2(i) are robust outexpanders is proved in Lemma 12.1 of [21]. Lemma 6.2 of [15] shows that the oriented graphs considered in Theorem 2(ii) are outexpanders. A similar proof shows that they are in fact robust outexpanders.…”

“…One could expect that for such graphs, a much weaker degree condition suffices. Indeed Häggkvist [12] pointed out that a minimum semi-degree of 3n− 4 8 is necessary and conjectured that it is also sufficient to guarantee a Hamilton cycle in any oriented graph of order n. The following approximate version of this conjecture was proved by Kelly, Kühn and Osthus [15].…”

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]). …”