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Embedding problems in graphs and hypergraphs

Treglown, Andrew Clark (2011)
Ph.D. thesis, University of Birmingham.

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Abstract

The first part of this thesis concerns perfect matchings and their generalisations. We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph, thereby answering a question of Hàn, Person and Schacht. We say that a graph \(G\) has a perfect \(H\)-packing (also called an \(H\) - factor) if there exists a set of disjoint copies of \(H\) in \(G\) which together cover all the vertices of \(G\). Given a graph \(H\), we determine, asymptotically, the Ore-type degree condition which ensures that a graph \(G\) has a perfect \(H\)-packing. The second part of the thesis concerns Hamilton cycles in directed graphs. We give a condition on the degree sequences of a digraph \(G\) that ensures \(G\) is Hamiltonian. This gives an approximate solution to a problem of Nash-Williams concerning a digraph analogue of Chvatal's theorem. We also show that every sufficiently large regular tournament can almost completely be decomposed into edge-disjoint Hamilton cycles. More precisely, for each \(\eta\) >0 every regular tournament \(G\) of sufficiently large order n contains at least (1/2- \(\eta\))n edge-disjoint Hamilton cycles. This gives an approximate solution to a conjecture of Kelly from 1968.

Type of Work:Ph.D. thesis.
Supervisor(s):Kuhn, Daniela and Osthus, Deryk
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:1345
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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