Extensions of Results on Rainbow Hamilton Cycles in Uniform Hypergraphs.

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 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)-bounded edge-colored $${K_n^{(k)}}$$ contains a properly colored ℓ-overlapping Hamilton cycle. [ABSTRACT FROM AUTHOR]