Abstract
Ford≥ 2, letHd(n,p)denote a randomd-uniform hypergraph withnvertices in which each of the$\binom{n}{d}$possible edges is present with probabilityp=p(n)independently, and letHd(n,m)denote a uniformly distributedd-uniform hypergraph withnvertices andmedges. Let eitherH=Hd(n,m)orH=Hd(n,p), wherem/nand$\binom{n-1}{d-1}p$need to be bounded away from (d−1)−1and 0 respectively. We determine the asymptotic probability thatHis connected. This yields the asymptotic number of connectedd-uniform hypergraphs with given numbers of vertices and edges. We also derive a local limit theorem for the number of edges inHd(n,p), conditioned onHd(n,p)being connected.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.