Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree

TL;DR

This work asymptotically determines the minimum degree threshold for forcing every orientation of a Hamilton cycle in large digraphs, showing that δ(G)≥(1+η)n suffices for all orientations except the directed cycle when G is not strongly connected. The authors develop a robust-outexpander partition framework that decomposes the graph into well-structured components and then embed any orientation via a two-case construction that hinges on long directed segments or switches within segments, aided by Taylor-type path results and universally k-linked connectivities. The results yield pancyclic variants and a directed 2-factor conclusion, extending and unifying classical theorems (e.g., Ghouila-Houri) and prior orientation results in tournaments and dense digraphs. An appendix provides the technical embedding toolkit, including splitting robust expanders, short/long path connections, and the linkage theorems, which are of independent interest for directed graph embeddings.

Abstract

In 1960, Ghouila-Houri proved that every strongly connected directed graph $G$ on $n$ vertices with minimum degree at least $n$ contains a directed Hamilton cycle. We asymptotically generalize this result by proving the following: every directed graph $G$ on $n$ vertices and with minimum degree at least $(1+o(1))n$ contains every orientation of a Hamilton cycle, except for the directed Hamilton cycle in the case when $G$ is not strongly connected. In fact, this minimum degree condition forces every orientation of a cycle in $G$ of every possible length, other than perhaps the directed cycles.

Figures (2)

• Figure 1: Case 1(b): An example with $q=4$
• Figure 2: Case 2, Step 2

Theorems & Definitions (30)

• Theorem 1.1: Ghouila-Houri GH
• Corollary 1.2: Ghouila-Houri GH
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5: DeBiasio, Kühn, Molla, Osthus and Taylor DKMOT
• Theorem 1.6
• Corollary 1.7
• Corollary 1.8
• Proposition 2.1
• Lemma 2.2