2024/09/30 by Yangyang Cheng, Mengjiao Rao, Cheng, Yangyang +5
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2409.20535
openalex publication_date 2024/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A 3-uniform loose cycle, denoted by Ct, is a 3-graph on t vertices whose vertices can be arranged cyclically so that each hyperedge consists of three consecutive vertices, and any two consecutive hyperedges share exactly one vertex. The length of Ct is the number of its hyperedges. We prove that for any η>0, there exists an n0=n0(η) such that for any n≥ n0 the following holds. Let C be a 3-graph consisting of vertex-disjoint loose cycles Cn1, Cn2, …, Cnr such that ∑i=1rni=n. Let k be the number of loose cycles with odd lengths in C. If H is a 3-graph on n vertices with minimum codegree at least (n+2k)/4+ηn, then H contains C as a spanning subhypergraph. The degree condition is approximately tight. This generalizes the result of Kühn and Osthus for loose Hamilton cycle and the result of Mycroft for loose cycle factors in 3-graphs. Our proof relies on the regularity lemma and a transversal blow-up lemma recently developed by the first author and Staden.