A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. 31 May 2016 / This dissertation examines the existing literature on set systems (or hypergraphs) and conducts an investigation of their (k-1)-overlapping diameters. In the general case of set systems of given diameter, some bounds on the possible sizes are given. We then restrict our focus to acyclic set systems and provide a full classification for simple, connected, acyclic, uniform set systems of each positive diameter, extending some results on trees. / GR 2016

Thesis (M.S.)--Rochester Institute of Technology, 1992. / Typescript. Includes bibliographical references.

Thesis (Ph.D)--Industrial and Systems Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Lee, Eva K.; Committee Member: Barnes, Earl; Committee Member: Johnson, Ellis; Committee Member: Parker, R. Gary; Committee Member: Wu, D. J.. Part of the SMARTech Electronic Thesis and Dissertation Collection.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

The class of squarefree monomial ideals is a classical object in commutative algebra, which has a strong connection to combinatorics. Our main goal throughout this dissertation is to study the algebraic properties of squarefree monomial ideals using combinatorial structures and invariants of hypergraphs. We focus on the following algebraic properties and invariants: the persistence property, non-increasing depth property, Castelnuovo-Mumford regularity and projective dimension. It has been believed for a long time that squarefree monomial ideals satisfy the persistence property and non-increasing depth property. In a recent work, Kaiser, Stehlik and Skrekovski provided a family of graphs and showed that the cover ideal of the smallest member of this family gives a counterexample to the persistence and non-increasing depth properties. We show that the cover ideals of all members of their family of graphs indeed fail to have the persistence and non-increasing depth properties. Castelnuovo-Mumford regularity and projective dimension are both important invariants in commutative algebra and algebraic geometry that govern the computational complexity of ideals and modules. Our focus is on finding bounds for the regularity in terms of combinatorial data from associated hypergraphs. We provide two upper bounds for the edge ideal of any vertex decomposable graph in terms of induced matching number and the number of cycles. We then give an upper bound for the edge ideal of a special class of vertex decomposable hypergraphs. Moreover, we generalize a domination parameter from graphs to hypergraphs and use it to give an upper bound for the projective dimension of the edge ideal of any hypergraph. / Mengyao Sun

This is an expository thesis. In this thesis we study out-orientations of hypergraphs, where every hyperarc has one tail vertex. We study hypergraphs that admit out-orientations covering supermodular-type connectivity requirements. For this, we follow a paper of Frank. We also study the Steiner rooted orientation problem. Given a hypergraph and a subset of vertices S ⊆ V, the goal is to give necessary and sufficient conditions for an orientation such that the connectivity between a root vertex and each vertex of S is at least k, for a positive integer k. We follow a paper by Kiraly and Lau, where they prove that every 2k-hyperedge connected hypergraph has such an orientation.

Thesis (Ph. D.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 85-90). Also available in print.

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