{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:19:30Z","timestamp":1759335570500,"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>A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is\u00a0linear\u00a0if it is simple and each pair of edges intersects in at most one vertex. For $n\\geq 3$, let $r= r(n)\\geq 3$ be an integer and let $\\boldsymbol{k} = (k_1,\\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \\sum_{j=1}^n k_j$ for all $n\\geq 3$, and define the set $\\mathcal{I} = \\{ n\\geq 3 \\mid r(n) \\text{ divides } M(n)\\}$. We assume that $\\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\\max}$ satisfies $r^4 k_{\\max}^4(k_{\\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.<\/jats:p>","DOI":"10.37236\/5512","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:40:31Z","timestamp":1578688831000},"source":"Crossref","is-referenced-by-count":4,"title":["Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees"],"prefix":"10.37236","volume":"23","author":[{"given":"Vladimir","family":"Blinovsky","sequence":"first","affiliation":[]},{"given":"Catherine","family":"Greenhill","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,8,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p17\/html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:17:54Z","timestamp":1579238274000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i3p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,8,5]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,7,8]]}},"URL":"https:\/\/doi.org\/10.37236\/5512","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,8,5]]},"article-number":"P3.17"}}