{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:02Z","timestamp":1753893842039,"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>Zarankiewicz's Conjecture (ZC) states that the crossing number\u00a0cr$(K_{m,n})$ equals $Z(m,n):=\\lfloor{\\frac{m}{2}}\\rfloor\u00a0\\lfloor{\\frac{m-1}{2}}\\rfloor\u00a0 \\lfloor{\\frac{n}{2}}\\rfloor \\lfloor{\\frac{n-1}{2}}\\rfloor$. Since\u00a0Kleitman's verification of ZC for $K_{5,n}$ (from which ZC for\u00a0$K_{6,n}$ easily follows), very little progress has been made around\u00a0ZC; the most notable exceptions involve computer-aided results. With\u00a0the aim of gaining a more profound understanding of this notoriously\u00a0difficult conjecture, we investigate the optimal (that is,\u00a0crossing-minimal) drawings of\u00a0$K_{5,n}$. The widely known natural drawings of $K_{m,n}$ (the\u00a0so-called Zarankiewicz drawings) with\u00a0$Z(m,n)$ crossings contain antipodal vertices, that is, pairs of\u00a0degree-$m$ vertices such that their induced drawing of $K_{m,2}$ has\u00a0no crossings. Antipodal vertices also play a major role in Kleitman's\u00a0inductive proof that cr$(K_{5,n}) = Z(5,n)$. We explore in depth\u00a0the role of antipodal vertices in optimal drawings of $K_{5,n}$, for\u00a0$n$ even. We prove that if\u00a0{$n \\equiv 2$ (mod $4$)}, then every optimal drawing\u00a0of $K_{5,n}$ has antipodal vertices. We also exhibit a two-parameter\u00a0family of optimal drawings $D_{r,s}$ of $K_{5,4(r+s)}$ (for $r,s\\ge 0$), with\u00a0no antipodal vertices, and show that if $n\\equiv 0$ (mod $4$), then\u00a0every optimal drawing of $K_{5,n}$ without\u00a0antipodal vertices is (vertex rotation) isomorphic to $D_{r,s}$ for some\u00a0integers $r,s$.\u00a0As a\u00a0corollary, we show that if $n$ is even, then\u00a0every optimal drawing of\u00a0$K_{5,n}$ is the superimposition of Zarankiewicz drawings with a\u00a0drawing isomorphic to $D_{r,s}$ for some nonnegative integers $r,s$.<\/jats:p>","DOI":"10.37236\/2777","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:54:33Z","timestamp":1578686073000},"source":"Crossref","is-referenced-by-count":7,"title":["The Optimal Drawings of $K_{5,n}$"],"prefix":"10.37236","volume":"21","author":[{"given":"C\u00e9sar","family":"Hern\u00e1ndez-V\u00e9lez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carolina","family":"Medina","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gelasio","family":"Salazar","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2014,10,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i4p1\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i4p1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:44:02Z","timestamp":1579239842000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i4p1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,2]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/2777","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,10,2]]},"article-number":"P4.1"}}