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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. Kuhn, Daniela and Osthus, Deryk Colleges (2008 onwards) > College of Engineering & Physical Sciences School of Mathematics QA Mathematics University of Birmingham 1345
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