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 (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 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 K (k) n 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 K (k) n 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. ∗Supported in part by NSF grant CCF1013110. †Supported in part by the Polish NSC grant N201 604940 and the NSF grant DMS-1102086. the electronic journal of combinatorics 19 (2012), #P46 1