{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T20:11:52Z","timestamp":1772136712678,"version":"3.50.1"},"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>Partial words are words that contain, in addition to letters, special symbols $\\diamondsuit$ called\u00a0holes. Two partial words $a=a_0 \\dots a_n$ and $b=b_0 \\dots b_n$ are compatible if, for all $i$, $a_i = b_i$ or at least one of $a_i, b_i$ is a hole. A partial word is unbordered if it does not have a proper nonempty prefix and a suffix that are compatible. Otherwise, the partial word is bordered.\r\nA set $R \\subseteq \\{0, \\dots, n\\}$ is called a complete sparse ruler of length $n$ if, for all $k \\in \\{0, \\dots, n\\}$, there exists $r, s \\in R$ such that $k = r - s$. These are also known as\u00a0restricted difference bases.From the definitions it follows that the more holes a partial word has, the more likely it is to be bordered. By introducing a connection between unbordered partial words and sparse rulers, we improve the bounds on the maximum number of holes an unbordered partial word can have over alphabets of sizes $4$ or greater. We also provide a counterexample for a previously reported theorem, depending on the reported values of the minimal number of marks in a length-$135$ ruler. We have not verified this value.We then study a two-dimensional generalization of these results. We adapt methods from the one-dimensional case to solve the correct asymptotic for the number of holes that an unbordered two-dimensional binary partial word can have.<\/jats:p>","DOI":"10.37236\/13806","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:34Z","timestamp":1772135014000},"source":"Crossref","is-referenced-by-count":0,"title":["A Connection Between Unbordered Partial Words and Sparse Rulers"],"prefix":"10.37236","volume":"33","author":[{"given":"Aleksi","family":"Saarela","sequence":"first","affiliation":[]},{"given":"Aleksi","family":"Vanhatalo","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,2,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p40\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p40\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:34Z","timestamp":1772135014000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p40"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,27]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/13806","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,27]]},"article-number":"P1.40"}}