Asymptotic enumeration of sparse uniform hypergraphs, with applications

dc.contributor.advisor Greenhill, Catherine en_US Aldosari, Haya en_US 2022-03-23T13:43:52Z 2022-03-23T13:43:52Z 2020 en_US
dc.description.abstract A hypergraph is a generalisation of a graph where the edges may contain more than two vertices. This thesis concentrates on asymptotic enumeration of some families of sparse uniform hypergraphs, when the number of vertices is sufficiently large and each vertex degree is given. Our first result gives an asymptotic formula for the number of simple uniform hypergraphs with a given degree sequence which contain no edges of another specified hypergraph. This formula holds under some restrictions on the number of forbidden edges, and the maximum degree and edge size of the hypergraph. We apply a combinatorial argument known as the switching method to obtain our estimates. This formula allows us to calculate the probability that a random uniform hypergraph with given degrees contains a specified set of edges. As a result, we find an asymptotic formula for the expected number of perfect matchings and the expected number of loose Hamilton cycles in random regular uniform hypergraphs. As another application of our first result, we study the average number of uniform spanning hypertrees. Finally, we give a new proof of Blinovsky and Greenhill's asymptotic formula for the number of sparse linear uniform hypergraphs with given degrees. Our proof uses a more complicated switching and extends the range of degrees and edge size for which the formula holds. en_US
dc.language English
dc.language.iso EN en_US
dc.publisher UNSW, Sydney en_US
dc.rights CC BY-NC-ND 3.0 en_US
dc.rights.uri en_US
dc.subject.other Spanning hypertree en_US
dc.subject.other Hypergraph en_US
dc.subject.other Linear hypergraph en_US
dc.subject.other Combinatorics en_US
dc.subject.other Hamilton cycle en_US
dc.subject.other Asymptotic enumeration en_US
dc.subject.other Perfect matching en_US
dc.title Asymptotic enumeration of sparse uniform hypergraphs, with applications en_US
dc.type Thesis en_US
dcterms.accessRights open access
dcterms.rightsHolder Aldosari, Haya
dspace.entity.type Publication en_US
unsw.relation.faculty Science
unsw.relation.originalPublicationAffiliation Aldosari, Haya, Mathematics & Statistics, Faculty of Science, UNSW en_US
unsw.relation.originalPublicationAffiliation Greenhill, Catherine, Mathematics & Statistics, Faculty of Science, UNSW en_US School of Mathematics & Statistics *
unsw.thesis.degreetype PhD Doctorate en_US
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