{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T12:45:01Z","timestamp":1759063501401,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A sequence is Thue or nonrepetitive if it does not contain a repetition of any length. We consider a generalization of this notion. A $j$-subsequence of a sequence $S$ is a subsequence in which two consecutive terms are at indices of difference $j$ in $S$. A $k$-Thue sequence is a sequence in which every $j$-subsequence, for $1\\le j \\le k$, is also Thue. It was conjectured that $k+2$ symbols are enough to construct an arbitrarily long $k$-Thue sequence and shown that the conjecture holds for $k \\in \\{2,3,5\\}$. In this paper we present a construction of $k$-Thue sequences on $2k$ symbols, which improves the previous bound of $2k + 10\\sqrt{k}$. Additionaly, we define cyclic $k$-Thue sequences and present a construction of such sequences of arbitrary lengths when $k=2$ using four symbols, with three exceptions. As a corollary, we obtain tight bounds for total Thue colorings of cycles. We conclude the paper with some open problems.<\/jats:p>","DOI":"10.37236\/4817","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T18:55:46Z","timestamp":1578682546000},"source":"Crossref","is-referenced-by-count":1,"title":["On a Generalization of Thue Sequences"],"prefix":"10.37236","volume":"22","author":[{"given":"Jaka","family":"Kranjc","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Borut","family":"Lu\u017ear","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martina","family":"Mockov\u010diakov\u00e1","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roman","family":"Sot\u00e1k","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2015,5,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p33\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i2p33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:16:57Z","timestamp":1579238217000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i2p33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,5,22]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2015,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/4817","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,5,22]]},"article-number":"P2.33"}}