2024/12/24 by Louis DeBiasio, DeBiasio, Louis, Jie Han +9 · 2 citations
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2412.18336
openalex publication_date 2024/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
The Pósa--Seymour conjecture determines the minimum degree threshold for forcing the kth 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 kth powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the kth 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.