Loose Hamilton Cycles in Regular Hypergraphs DudekAndrzej FriezeAlan RucinskiAndrzej SileikisMatas 2013 <p>We establish a relation between two uniform models of random <em>k</em>-graphs (for constant <em>k</em> ≥ 3) on <em>n</em> labelled vertices: ℍ<sup><em>(k)</em> </sup><em>(n,m)</em>, the random <em>k</em>-graph with exactly <em>m</em> edges, and ℍ<sup> <em>(k)</em> </sup><em>(n,d)</em>, the random <em>d</em>-regular <em>k</em>-graph. By extending the switching technique of McKay and Wormald to <em>k</em>-graphs, we show that, for some range of <em>d = d(n)</em> and a constant <em>c</em>> 0, if <em>m</em> ~ <em>cnd</em>, then one can couple ℍ<sup> <em>(k)</em> </sup><em>(n,m)</em> and ℍ<sup> <em>(k)</em> </sup><em>(n,d)</em> so that the latter contains the former with probability tending to one as <em>n</em> → ∞. In view of known results on the existence of a loose Hamilton cycle in ℍ<sup> <em>(k)</em> </sup><em>(n,m)</em>, we conclude that ℍ<sup> <em>(k)</em> </sup><em>(n,d)</em> contains a loose Hamilton cycle when <em>d</em> ≫ log <em>n</em> (or just <em>d</em> ≥ <em>C</em> log <em>n</em>, if <em>k</em> = 3) and <em>d</em> = <em>o</em>(<em>n</em><sup>1/2</sup>).</p>