{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:18:48Z","timestamp":1758824328550,"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 tight crossing number inequalities for geometric graphs whose\u00a0vertex sets are taken from a $d$-dimensional grid of volume $N$\u00a0and give applications of these inequalities to counting the number\u00a0of crossing-free geometric graphs that can be drawn on such grids.In particular, we show that any geometric graph with $m\\geq 8N$ edges and with vertices on a 3D integer grid of volume $N$, has $\\Omega((m^2\/N)\\log(m\/N))$ crossings. In $d$-dimensions, with $d\\ge 4$, this bound becomes $\\Omega(m^2\/N)$. We provide matching upper bounds for all $d$. Finally, for $d\\ge 4$ the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some $d$-dimensional grid of volume $N$ is $N^{\\Theta(N)}$. In 3 dimensions it remains open to improve the trivial bounds, namely, the $2^{\\Omega(N)}$ lower bound and the $N^{O(N)}$ upper bound.<\/jats:p>","DOI":"10.37236\/3025","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:08:58Z","timestamp":1578686938000},"source":"Crossref","is-referenced-by-count":3,"title":["Crossings in Grid Drawings"],"prefix":"10.37236","volume":"21","author":[{"given":"Vida","family":"Dujmovi\u0107","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pat","family":"Morin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adam","family":"Sheffer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2014,2,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p41\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i1p41\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T06:04:53Z","timestamp":1579241093000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i1p41"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,1,13]]}},"URL":"https:\/\/doi.org\/10.37236\/3025","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2014,2,28]]},"article-number":"P1.41"}}