Loose Hamilton Cycles in Regular Hypergraphs. (24th September 2014) - DUDEK, ANDRZEJ; FRIEZE, ALAN; RUCIŃSKI, ANDRZEJ; ŠILEIKIS, MATAS
Abstract : We establish a relation between two uniform models of random k -graphs (for constant k ⩾ 3) on n labelled vertices: ℍ (k) (n, m), the random k -graph with exactly m edges, and ℍ (k) (n, d), the random d -regular k -graph. By extending the switching technique of McKay and Wormald to k -graphs, we show that, for some range of d = d(n) and a constant c > 0, if m ~ cnd, then one can couple ℍ (k) (n, m) and ℍ (k) (n, d) so that the latter contains the former with probability tending to one as n → ∞. In view of known results on the existence of a loose Hamilton cycle in ℍ (k) (n, m), we conclude that ℍ (k) (n, d) contains a loose Hamilton cycle when d ≫ log n (or just d ⩾ C log n, if k = 3) and d = o ( n 1/2 ).