Arbitrary Orientations of Hamilton Cycles in Oriented Graphs

Abstract

We use a randomised embedding method to prove that for all α > 0 any sufficiently large oriented graph G with minimum in-degree and out-degree δ + (G), δ − (G) ≥ (3/8 + α)|G| contains every possible orientation of a Hamilton cycle. This confirms a conjecture of Häggkvist and Thomason.

Arbitrary Orientations of Hamilton Cycles in Digraphs

“…In Lemma 4.4, we observe that any graph satisfying the conditions of Theorem 1.2 must be a robust outexpander or have a large set which does not expand, in which case we say that G is ε-extremal. Theorem 1.2 was verified for the case when G is a robust outexpander by Taylor [13] based on the approach of Kelly [9]. This allows us to restrict our attention to the ε-extremal case.…”

“…Recall from Section 1 that Kelly [9] showed that any sufficiently large oriented graph with minimum semidegree at least (3/8 + α)n contains any orientation of a Hamilton cycle. It is not hard to show that any such oriented graph is a robust outexpander (see [11]).…”

“…It is not hard to show that any such oriented graph is a robust outexpander (see [11]). In fact, in [9], Kelly observed that his arguments carry over to robustly expanding digraphs of linear degree. Taylor [13] has verified that this is indeed the case, proving the following result.…”

“…Kelly [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).…”

A survey on Hamilton cycles in directed graphs

“…Theorem 25 (Kelly [39]). For every α > 0 there exists an integer n 0 = n 0 (α) such that every oriented graph G on n ≥ n 0 vertices with minimum semidegree δ 0 (G) ≥ (3/8 + α)n contains every orientation of a Hamilton cycle.…”