Extensions of Results on Rainbow Hamilton Cycles in Uniform Hypergraphs

Let K (k) n be the complete k-uniform hypergraph, k ≥ 3, and let ` be an integer such that 1 ≤ ` ≤ k − 1 and k − ` divides n. An `-overlapping Hamilton cycle in K n is a spanning subhypergraph C of K (k) n 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 consecutive edges in C intersects in precisely ` vertices. An edge-coloring of K (k) n is (a, r)-bounded if every subset of a vertices of K (k) n is contained in at most r edges of the same color. In this paper, we refine recent results of the first author, Frieze and Ruciński by proving that there is a constant c = c(k, `) such that every (`, cnk−`)-bounded edge-colored K (k) n in which no color appears more that cnk−1 times contains a rainbow `-overlapping Hamilton cycle. We also show that there is a constant c′ = c′(k, `) such that every (`, c′nk−`)-bounded edge-colored K (k) n contains a properly colored `-overlapping Hamilton cycle.