{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:56Z","timestamp":1753893836790,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We define a $d$-balanced equi-$n$-square $L=(l_{ij})$, for some divisor $d$ of $n$, as an $n \\times n$ matrix containing symbols from $\\mathbb{Z}_n$ in which any symbol that occurs in a row or column, occurs exactly $d$ times in that row or column. We show how to construct a $d$-balanced equi-$n$-square from a partition of a Latin square of order $n$ into $d \\times (n\/d)$ subrectangles. In graph theory, $L$ is equivalent to a decomposition of $K_{n,n}$ into $d$-regular spanning subgraphs of $K_{n\/d,n\/d}$. We also study when $L$ is diagonally cyclic, defined as when $l_{(i+1)(j+1)}=l_{ij}+1$ for all $i,j \\in \\mathbb{Z}_n$, which correspond to cyclic such decompositions of $K_{n,n}$ (and thus $\\alpha$-labellings).\r\nWe identify necessary conditions for the existence of (a) $d$-balanced equi-$n$-squares, (b) diagonally cyclic $d$-balanced equi-$n$-squares, and (c) Latin squares of order $n$ which partition into $d \\times (n\/d)$ subrectangles. We prove the necessary conditions are sufficient for arbitrary fixed $d \\geq 1$ when $n$ is sufficiently large, and we resolve the existence problem completely when $d \\in \\{1,2,3\\}$.\r\nAlong the way, we identify a bijection between $\\alpha$-labellings of $d$-regular bipartite graphs and what we call $d$-starters: matrices with exactly one filled cell in each top-left-to-bottom-right unbroken diagonal, and either $d$ or $0$ filled cells in each row and column. We use $d$-starters to construct diagonally cyclic $d$-balanced equi-$n$-squares, but this also gives new constructions of $\\alpha$-labellings.<\/jats:p>","DOI":"10.37236\/9118","type":"journal-article","created":{"date-parts":[[2020,10,21]],"date-time":"2020-10-21T04:05:36Z","timestamp":1603253136000},"source":"Crossref","is-referenced-by-count":0,"title":["Balanced Equi-$n$-Squares"],"prefix":"10.37236","volume":"27","author":[{"given":"Saieed","family":"Akbari","sequence":"first","affiliation":[]},{"given":"Trent G.","family":"Marbach","sequence":"additional","affiliation":[]},{"given":"Rebecca J.","family":"Stones","sequence":"additional","affiliation":[]},{"given":"Zhuanhao","family":"Wu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,10,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p8\/8189","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p8\/8189","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,21]],"date-time":"2020-10-21T04:05:36Z","timestamp":1603253136000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i4p8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,16]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/9118","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,10,16]]},"article-number":"P4.8"}}