Powers of Hamilton cycles in oriented and directed graphs

TL;DR

This paper determines the minimum degree thresholds needed to guarantee the existence of powers of Hamilton cycles in oriented and directed graphs, extending classical conjectures and providing new bounds.

Contribution

It asymptotically establishes the minimum total degree threshold for the square of a Hamilton cycle in digraphs and proposes a conjecture for higher powers, also offering a semi-degree condition for oriented graphs.

Findings

Established the threshold for the square of a Hamilton cycle in digraphs.

Proposed a conjecture for the thresholds of higher powers.

Provided a semi-degree condition for oriented graphs that captures the correct order of magnitude.

Abstract

The P\'osa--Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Koml\'os, S\'ark\"ozy and Szemer\'edi 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. Tur\'an-type problems for oriented graphs are also discussed.