Rainbow Hamilton Cycles in Uniform Hypergraphs

Let Kₙ⁽ᵏ⁾ 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 Kₙ⁽ᵏ⁾ is a spanning subhypergraph C of Kₙ⁽ᵏ⁾ 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.We show that, for some constant c=c(k,) and sufficiently large n, for every coloring (partition) of the edges of Kₙ⁽ᵏ⁾ which uses arbitrarily many colors but no color appears more than cnᵏ⁻ times, there exists a rainbow -overlapping Hamilton cycle C, that is every edge of C receives a different color. We also prove that, for some constant c'=c'(k,) and sufficiently large n, for every coloring of the edges of Kₙ⁽ᵏ⁾ in which the maximum degree of the subhypergraph induced by any single color is bounded by c'nᵏ⁻, there exists a properly colored -overlapping Hamilton cycle C, that is every two adjacent edges receive different colors. For =1, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for 2≤≤ k-1.The proofs rely on a version of the Lovász Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.