Extensions of Results on Rainbow Hamilton Cycles in Uniform Hypergraphs

Abstract

Let ${K_n^{(k)}}$ be the complete k-uniform hypergraph, ${k\ge3}$, and let ℓ be an integer such that 1 ≤ ℓ ≤ k−1 and k−ℓ divides n. An ℓ-overlapping Hamilton cycle in ${K_n^{(k)}}$ is a spanning subhypergraph C of ${K_n^{(k)}}$ 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 consecutive edges in C intersects in precisely ℓ vertices. An edge-coloring of ${K_n^{(k)}}$ is (a, r)-bounded if every subset of a vertices of ${K_n^{(k)}}$ 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, ℓ) such that every ${(\ell, cn^{k-\ell})}$ -bounded edge-colored ${K_n^{(k)}}$ in which no color appears more that cn ^k-1 times contains a rainbow ℓ-overlapping Hamilton cycle. We also show that there is a constant c′ = c′(k, ℓ) such that every (ℓ, c′n ^k-ℓ)-bounded edge-colored ${K_n^{(k)}}$ contains a properly colored ℓ-overlapping Hamilton cycle.