Arbitrary orientations of cycles in oriented graphs

TL;DR

The paper resolves Häggkvist and Thomason's long-standing question by showing that for all large $n$ an oriented graph with minimum semidegree at least $\frac{3n-1}{8}$ contains every possible orientation of a Hamilton cycle, with a tightened bound $\frac{3n-4}{8}$ for nearly directed cycles. The authors leverage a robust-outexpander framework, a delta-extremal partition $(W,X,Y,Z)$, and a combination of probabilistic embedding, the Diregularity Lemma, and the Blow-up Lemma to realize embeddings in both expansion and near-extremal regimes. They split the analysis into two main cases based on sinks in the Hamilton cycle: when $C$ has few sinks, and when $C$ has many sinks, developing two tailored embedding strategies that balance clusters and exploit special edges, respectively. A pancyclicity extension shows that the same threshold yields cycles of every orientation and every length, except for a small class of exceptional graphs. Overall, the work advances exact threshold results for Hamiltonian orientations in oriented graphs and strengthens the toolkit for embedding complex structures via regularity-based methods, with potential transversal generalizations to families of digraphs.

Abstract

We show that every sufficiently large oriented graph $G$ with minimum indegree and outdegree both at least $(3|V(G)|-1)/8$ contains every orientation of a Hamilton cycle. This result improves the approximate bound established by Kelly and resolves a long-standing problem posed by Häggkvist and Thomason in 1995. The degree condition is tight and it can be improved to $(3|V(G)|-4)/8$ for Hamilton cycles that are nearly directed, generalizing a classic result by Keevash, Kühn and Osthus. Additionally, we derive a pancyclicity result for arbitrary orientations. More precisely, the above degree condition suffices to guarantee the existence of cycles of every possible orientation and every possible length unless $G$ is isomorphic to one of the exceptional oriented graphs.

Figures (4)

• Figure 1: The oriented graph $G$ in Proposition \ref{['PROP-degreesharp']}. The bold edges indicate that all possible edges are present and have the directed shown. The set $W$ has size $\lfloor n/4\rfloor$ and $|X|=|Z|=\lceil n/4\rceil$. Each of $W$ and $Y$ spans an almost regular tournament, that is, the indegree and outdegree of every vertex differ by at most one. The oriented graph induced by $X$ and $Z$ is an almost regular bipartite tournament. Table \ref{['TAB-degreesharp']} in Appendix \ref{['APPSEC-table1']} will help readers to better check every vertex of $G$ has the correct indegree and outdegree.
• Figure 2: An illustration of how to embed $C$ when it contains few sinks. The black diamonds, white squares and black circles indicate the sources, sinks and normal vertices of $C$, respectively. The white circle on the path $L_2$ in (b) indicates that it is a bad vertex of $G$. In the proof we use directed 13-paths to cover bad vertices but here we use a 5-path $L_2$ for an illustration. Note that the blue fat directed path $R_1$ has the same direction as the cycle $C$ but the red dashed directed path $R_2$ has the opposite direction.
• Figure 3: An illustration of how randomised embedding method may be applied to $G$.
• Figure 4: An illustration of how to cover all vertices in $W^\prime\backslash(B\cup U)$ by embedding $P_W$. The path in $G[W^\prime\backslash(B\cup U)]$ is the auxiliary path $P_W^{\ast}$ obtained from $P_W$ by replacing several short subpaths with edges.

Theorems & Definitions (73)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3: kellyJCTB100
• Theorem 1.4: kellyEJC18
• Theorem 1.5
• Theorem 4.1: Chernoff Bound 1
• Lemma 4.2: debiasioEJC22
• Theorem 4.3: Chernoff Bound 2
• Lemma 4.4
• proof