{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T13:42:05Z","timestamp":1769089325683,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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\u00a0 $K_n^{(k)}$\u00a0 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 $c'n^{k-\\ell}$,\u00a0 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\u00a0 rely on a version of the Lov\u00e1sz Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.<\/jats:p>","DOI":"10.37236\/2055","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:31:23Z","timestamp":1578713483000},"source":"Crossref","is-referenced-by-count":18,"title":["Rainbow Hamilton Cycles in Uniform Hypergraphs"],"prefix":"10.37236","volume":"19","author":[{"given":"Andrzej","family":"Dudek","sequence":"first","affiliation":[]},{"given":"Alan","family":"Frieze","sequence":"additional","affiliation":[]},{"given":"Andrzej","family":"Ruci\u0144ski","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2012,2,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p46\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v19i1p46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T22:41:38Z","timestamp":1579300898000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v19i1p46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2012,2,15]]}},"URL":"https:\/\/doi.org\/10.37236\/2055","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,2,23]]},"article-number":"P46"}}