Asymptotic enumeration of sparse uniform hypergraphs with given degrees

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• 作者：Vladimir Blinovsky^a ; ^b ; ^{vblinovs@yandex.ru" class="auth_mail" title="E-mail the corresponding author} ; Catherine Greenhill^c ; ^{c.greenhill@unsw.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：51
• 期：Complete
• 页码：287-296
• 全文大小：402 K

文摘

Let 95669815001389&_mathId=si20.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=eb629daa589822ec4c90b9555f153216" title="Click to view the MathML source">r≥2$r\ge 2$ be a fixed integer. For infinitely many 95669815001389&_mathId=si37.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=68d88af629f91651d0daeef9952744d4" title="Click to view the MathML source">n$n$, let 95669815001389&_mathId=si4.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=683bc9e5bb3a27a455672d3e848e5e9a">95669815001389-si4.gif">$k=(k_1,\dots ,k_n)$ be a vector of nonnegative integers such that their sum 95669815001389&_mathId=si207.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=3c4a0840be561a3e95b9d9abb7fba674" title="Click to view the MathML source">M$M$ is divisible by 95669815001389&_mathId=si104.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=27768e51bbc7f5dba645d8b835c66400" title="Click to view the MathML source">r$r$. We present an asymptotic enumeration formula for simple 95669815001389&_mathId=si104.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=27768e51bbc7f5dba645d8b835c66400" title="Click to view the MathML source">r$r$-uniform hypergraphs with degree sequence 95669815001389&_mathId=si119.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=b60f72dd76b02441ad8f79812ab1ba59">95669815001389-si119.gif">$k$. (Here “simple” means that all edges are distinct and no edge contains a repeated vertex.) Our formula holds whenever the maximum degree 95669815001389&_mathId=si55.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=605e2210ef888c4153c43f1b66ead013">95669815001389-si55.gif">$k_{\max }$ satisfies 95669815001389&_mathId=si10.gif&_user=111111111&_pii=S0195669815001389&_rdoc=1&_issn=01956698&md5=7178a124b743f41b7ad0781c1683fe51">95669815001389-si10.gif">$k_{\max }^3=o\left(M\right)$.