{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:21:28Z","timestamp":1759335688075,"version":"3.41.2"},"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>We prove that each maximal partial Latin cube must have more than $29.289\\%$ of its cells filled and show by construction that this is a nearly tight bound. We also prove upper and lower bounds on the number of cells containing a fixed symbol in maximal partial Latin cubes and hypercubes, and we use these bounds to determine for small orders $n$ the numbers $k$ for which there exists a maximal partial Latin cube of order $n$ with exactly $k$ entries. Finally, we prove that maximal partial Latin cubes of order $n$ exist of each size from approximately half-full ($n^3\/2$ for even $n\\geq 10$ and $(n^3+n)\/2$ for odd $n\\geq 21$) to completely full, except for when either precisely $1$ or $2$ cells are empty.<\/jats:p>","DOI":"10.37236\/4726","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:22:20Z","timestamp":1578702140000},"source":"Crossref","is-referenced-by-count":4,"title":["Maximal Partial Latin Cubes"],"prefix":"10.37236","volume":"22","author":[{"given":"Thomas","family":"Britz","sequence":"first","affiliation":[]},{"given":"Nicholas J.","family":"Cavenagh","sequence":"additional","affiliation":[]},{"given":"Henrik Kragh","family":"S\u00f8rensen","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,3,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p81\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p81\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:23:13Z","timestamp":1579256593000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p81"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4726","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,3,30]]},"article-number":"P1.81"}}