dc.contributor.author Sato, Cristiane Maria dc.date.accessioned 2013-08-30 15:21:48 (GMT) dc.date.available 2013-08-30 15:21:48 (GMT) dc.date.issued 2013-08-30T15:21:48Z dc.date.submitted 2013 dc.identifier.uri http://hdl.handle.net/10012/7787 dc.description.abstract The k-core of a graph is its maximal subgraph with minimum degree at least k. The study of k-cores in random graphs was initiated by Bollobás in 1984 in connection to k-connected subgraphs of random graphs. Subsequently, k-cores and their properties have been extensively investigated in random graphs and hypergraphs, with the determination of the threshold for the emergence of a giant k-core, due to Pittel, Spencer and Wormald, as one of the most prominent results. en In this thesis, we obtain an asymptotic formula for the number of 2-connected graphs, as well as 2-edge-connected graphs, with given number of vertices and edges in the sparse range by exploiting properties of random 2-cores. Our results essentially cover the whole range for which asymptotic formulae were not described before. This is joint work with G. Kemkes and N. Wormald. By defining and analysing a core-type structure for uniform hypergraphs, we obtain an asymptotic formula for the number of connected 3-uniform hypergraphs with given number of vertices and edges in a sparse range. This is joint work with N. Wormald. We also examine robustness aspects of k-cores of random graphs. More specifically, we investigate the effect that the deletion of a random edge has in the k-core as follows: we delete a random edge from the k-core, obtain the k-core of the resulting graph, and compare its order with the original k-core. For this investigation we obtain results for the giant k-core for Erdős-Rényi random graphs as well as for random graphs with minimum degree at least k and given number of vertices and edges. dc.language.iso en en dc.publisher University of Waterloo en dc.subject combinatorics en dc.subject graph theory en dc.subject random graphs en dc.subject probabilistic en dc.subject enumeration en dc.title Core Structures in Random Graphs and Hypergraphs en dc.type Doctoral Thesis en dc.pending false en dc.subject.program Combinatorics and Optimization en uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree Doctor of Philosophy en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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