{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T01:32:58Z","timestamp":1783733578791,"version":"3.55.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A graph is called $1$-planar if it admits a drawing in the plane such that each edge is crossed at most once. Let $G$ be a bipartite $1$-planar graph with $n$ ($n\\ge 4$) vertices and $m$ edges. Karpov showed that $m\\le 3n-8$ holds for even $n\\ge 8$ and $m\\le 3n-9$ holds for odd $n\\ge 7$. Czap, Przyby\u0142o and \u0160krabul'\u00e1kov\u00e1 proved that if the partite sets of $G$ are of sizes $x$ and $y$, then $m\\le 2n+6x-12$ holds for $2\\leq x\\leq y$, and conjectured that $m\\le 2n+4x-12$ holds for $x\\ge 3$ and $y\\ge 6x-12$. In this paper, we settle their conjecture and our result is even under a weaker condition $2\\le x\\le y$.<\/jats:p>","DOI":"10.37236\/10012","type":"journal-article","created":{"date-parts":[[2021,5,21]],"date-time":"2021-05-21T03:11:22Z","timestamp":1621566682000},"source":"Crossref","is-referenced-by-count":2,"title":["On the Sizes of Bipartite 1-Planar Graphs"],"prefix":"10.37236","volume":"28","author":[{"given":"Yuanqiu","family":"Huang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhangdong","family":"Ouyang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fengming","family":"Dong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2021,5,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i2p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i2p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,21]],"date-time":"2021-05-21T03:11:23Z","timestamp":1621566683000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i2p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,21]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/10012","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,21]]},"article-number":"P2.22"}}