Loose Hamilton Cycles in Regular Hypergraphs

• Published in: arXiv.org
• Main Authors: Dudek, Andrzej, Frieze, Alan, Ruciński, Andrzej, Matas Šileikis
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.04.2013
• Subjects: Apexes; Graph theory; Graphs; Mathematics - Combinatorics
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1304.1426

We establish a relation between two uniform models of random \(k\)-graphs (for constant \(k \ge 3\)) on \(n\) labeled vertices: \(H(n,m)\), the random \(k\)-graph with exactly \(m\) edges, and \(H(n,d)\), the random \(d\)-regular \(k\)-graph. By extending to \(k\)-graphs the switching technique of McKay and Wormald, we show that, for some range of \(d = d(n)\) and a constant \(c > 0\), if \(m \sim cnd\), then one can couple \(H(n,m)\) and \(H(n,d)\) so that the latter contains the former with probability tending to one as \(n \to \infty\). In view of known results on the existence of a loose Hamilton cycle in \(H(n,m)\), we conclude that \(H(n,d)\) contains a loose Hamilton cycle when \(\log n = o(d)\) (or just \(d \ge C log n\), if \(k = 3\)) and \(d = o(n^{1/2})\).