On rainbow Hamilton cycles in random hypergraphs

Let H-n,p,r((k)) 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 [r]. Let H be a k-uniform hypergraph of order n. An l-Hamilton cycle is a spanning subhypergraph C of H with n/(k - l) 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 l vertices. In this note we study the existence of rainbow l-Hamilton cycles (that is every edge receives a different color) in H-n,p,r((k)). We mainly focus on the most restrictive case when r = n/(k In particular, we show that for the so called tight Hamilton cycles (l = k 1) p = e(2)/n is the sharp threshold for the existence of a rainbow tight Hamilton cycle in H-n,p,r((k)) for each k >= 4.