{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:14Z","timestamp":1753893794893,"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>Suppose $p &lt; q$ are odd and relatively prime. In this paper we complete the proof that $K_{n,n}$ has a factorisation into factors $F$ whose components are copies of $K_{p,q}$ if and only if $n$ is a multiple of $pq(p+q)$. The final step is to solve the \"c-value problem\" of Martin. This is accomplished by proving the following fact and some variants: For any $0\\le k\\le n$, there exists a sequence $(\\pi_1,\\pi_2, \\dots,\\pi_{2k+1})$ of (not necessarily distinct) permutations of $\\{1,2,\\dots,n\\}$ such that each value in $\\{-k,1-k,\\dots,k\\}$ occurs exactly $n$ times as $\\pi_j(i)-i$ for $1\\le j\\le 2k-1$ and $1\\le i\\le n$.<\/jats:p>","DOI":"10.37236\/1730","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T02:03:12Z","timestamp":1578621792000},"source":"Crossref","is-referenced-by-count":4,"title":["Permutation Separations and Complete Bipartite Factorisations of $K_{n,n}$"],"prefix":"10.37236","volume":"10","author":[{"given":"Nigel","family":"Martin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richard","family":"Stong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2003,9,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r37\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v10i1r37\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:06:49Z","timestamp":1579306009000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v10i1r37"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,9,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2003,1,6]]}},"URL":"https:\/\/doi.org\/10.37236\/1730","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2003,9,17]]},"article-number":"R37"}}