Rainbow Hamilton Cycles in Uniform Hypergraphs

Abstract

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, $k\ge3$, and let $\ell$ be an integer such that $1\le \ell\le k-1$ and $k-\ell$ divides $n$. An $\ell$-overlapping Hamilton cycle in $K_n^{(k)}$ is a spanning subhypergraph $C$ of  $K_n^{(k)}$  with $n/(k-\ell)$ 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 $\ell$ vertices.We show that, for some constant $c=c(k,\ell)$ and sufficiently large $n$, for every coloring (partition) of the edges of $K_n^{(k)}$ which uses arbitrarily many colors but no color appears more than $cn^{k-\ell}$ times, there exists a rainbow $\ell$-overlapping Hamilton cycle $C$, that is every edge of $C$ receives a different color. We also prove that, for some constant $c'=c'(k,\ell)$ and sufficiently large $n$, for every coloring of the edges of $K_n^{(k)}$ in which the maximum degree of the subhypergraph induced by any single color is bounded by $c'n^{k-\ell}$,  there exists a properly colored $\ell$-overlapping Hamilton cycle $C$, that is every two adjacent edges receive different colors. For $\ell=1$, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for $2\le\ell\le k-1$.The proofs  rely on a version of the Lovász Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.