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Powers of Hamilton cycles in oriented and directed graphs

Louis DeBiasio, Jie Han, Allan Lo, Theodore Molla, Simón Piga, Andrew Treglown

TL;DR

This work extends the theory of Hamilton-cycle powers to digraphs and oriented graphs by establishing asymptotically tight minimum-degree conditions for the square of a Hamilton cycle in digraphs and near-optimal semi-degree thresholds in oriented graphs for general powers. The authors develop a directed absorption–connecting framework, supported by the diregularity and blow-up lemmas, to assemble a spanning $C^k$ via an absorbing path, a reservoir for connections, and an almost-covering stage that is then finalized by absorption. They provide extremal constructions showing near-tightness, conjecture asymptotic thresholds for general $k$, and discuss Turán-type and tiling problems in oriented graphs, including exact results for Turánability of powers of cycles. The results illuminate the interplay between degree conditions, connectivity, and structured embeddings and have potential implications for bandwidth-type results and tiling in digraphs and oriented graphs. The paper also surveys open problems and outlines a clear path toward extending the asymptotic results to wider ranges of $k$.

Abstract

The Pósa--Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Komlós, Sárközy and Szemerédi proved the conjecture for sufficiently large graphs. In this paper we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for $k$th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the $k$th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Turán-type problems for oriented graphs are also discussed.

Powers of Hamilton cycles in oriented and directed graphs

TL;DR

This work extends the theory of Hamilton-cycle powers to digraphs and oriented graphs by establishing asymptotically tight minimum-degree conditions for the square of a Hamilton cycle in digraphs and near-optimal semi-degree thresholds in oriented graphs for general powers. The authors develop a directed absorption–connecting framework, supported by the diregularity and blow-up lemmas, to assemble a spanning via an absorbing path, a reservoir for connections, and an almost-covering stage that is then finalized by absorption. They provide extremal constructions showing near-tightness, conjecture asymptotic thresholds for general , and discuss Turán-type and tiling problems in oriented graphs, including exact results for Turánability of powers of cycles. The results illuminate the interplay between degree conditions, connectivity, and structured embeddings and have potential implications for bandwidth-type results and tiling in digraphs and oriented graphs. The paper also surveys open problems and outlines a clear path toward extending the asymptotic results to wider ranges of .

Abstract

The Pósa--Seymour conjecture determines the minimum degree threshold for forcing the th power of a Hamilton cycle in a graph. After numerous partial results, Komlós, Sárközy and Szemerédi proved the conjecture for sufficiently large graphs. In this paper we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Turán-type problems for oriented graphs are also discussed.

Paper Structure

This paper contains 25 sections, 34 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Powers of Hamilton cycles in digraphs
  3. Powers of Hamilton cycles in oriented graphs
  4. Overview of the proofs of Theorems \ref{['thm:square']} and \ref{['thm:main']}
  5. The extremal examples
  6. The extremal example for Conjecture \ref{['conjnew']}
  7. The extremal example for Theorem \ref{['thm:main']}
  8. The regularity lemma and related results
  9. Directed graphs: almost covering, absorbing, and connecting lemmas for Theorem \ref{['thm:square']}
  10. Almost Covering Lemma
  11. Absorbing Lemma
  12. Connecting Lemma
  13. Oriented graphs: absorbing, connecting, and almost covering lemmas for Theorem \ref{['thm:square']}
  14. Auxiliary lemmas for connecting and absorbing
  15. The absorbing lemma
  16. ...and 10 more sections

Key Result

Theorem 1.4

Given any $\eta>0$, there exists $n_0 \in \mathbb N$ so that for any $n \geq n_0$ the following holds. If $G$ is an $n$-vertex digraph with then $G$ contains the square of a Hamilton cycle.

Figures (1)

  • Figure 1: The oriented graph $G_{2}$ does not contain a square of Hamilton cycle.

Theorems & Definitions (73)

  • Conjecture 1.1: Pósa and Seymour, see posaseymour
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 63 more