{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:57Z","timestamp":1753893777790,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The binomial random bipartite graph $G(n,n,p)$ is the random graph formed by taking two partition classes of size $n$ and including each edge between them independently with probability $p$. It is known that this model exhibits a similar phase transition as that of the binomial random graph $G(n,p)$ as $p$ passes the critical point of $\\frac{1}{n}$. We study the component structure of this model near to the critical point. We show that, as with $G(n,p)$, for an appropriate range of $p$ there is a unique `giant' component and we determine asymptotically its order and excess. We also give more precise results for the distribution of the number of components of a fixed order in this range of $p$. These results rely on new bounds for the number of bipartite graphs with a fixed number of vertices and edges, which we also derive.<\/jats:p>","DOI":"10.37236\/11065","type":"journal-article","created":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T09:23:17Z","timestamp":1689326597000},"source":"Crossref","is-referenced-by-count":0,"title":["Component Behaviour and Excess of Random Bipartite Graphs Near the Critical Point"],"prefix":"10.37236","volume":"30","author":[{"given":"Tuan","family":"Do","sequence":"first","affiliation":[]},{"given":"Joshua","family":"Erde","sequence":"additional","affiliation":[]},{"given":"Mihyun","family":"Kang","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Missethan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,7,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i3p7\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i3p7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T09:23:34Z","timestamp":1689326614000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i3p7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,14]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,7,14]]}},"URL":"https:\/\/doi.org\/10.37236\/11065","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,7,14]]},"article-number":"P3.7"}}