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## Embedding problems in graphs and hypergraphs

### 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

Topics: QA Mathematics
Year: 2011
OAI identifier: oai:etheses.bham.ac.uk:1345

### Citations

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