On Rainbow Hamilton Cycles in Random Hypergraphs

Let H (k) n,p,κ denote a randomly colored random hypergraph, constructed on the vertex set [n] by taking each k-tuple independently with probability p, and then independently coloring it with a random color from the set [κ]. Let H be a k-uniform hypergraph of order n. An `-Hamilton cycle is a spanning subhypergraph C of H with n/(k−`) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of adjacent edges in C intersects in precisely ` vertices. In this note we study the existence of rainbow `-Hamilton cycles (that is every edge receives a different color) in H (k) n,p,κ. We mainly focus on the most restrictive case when κ = n/(k−`). In particular, we show that for the so called tight Hamilton cycles (` = k−1) p = e/n is the sharp threshold for the existence of a rainbow tight Hamilton cycle in H (k) n,p,n for each k ≥ 4.