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Title: Embedding spanning structures in graphs and hypergraphs
Author: Knox, Fiachra
ISNI:       0000 0004 2730 5517
Awarding Body: University of Birmingham
Current Institution: University of Birmingham
Date of Award: 2013
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In this thesis we prove three main results on embeddings of spanning subgraphs into graphs and hypergraphs. The first is that for log⁵⁰ n/n \ ≤ p ≤ 1-n⁻¹/⁴ log⁹ n, a binomial random graph G ~ G_n,p contains with high probability a collection of └δ(G)/2┘ edge disjoint Hamilton cycles (plus an additional edge-disjoint matching if δ(G) is odd), which confirms for this range of p a conjecture of Frieze and Krivelevich. Secondly, we show that any 'robustly expanding' graph with linear minimum degree on sufficiently many vertices contains every bipartite graph on the same number of vertices with bounded maximum degree and sublinear bandwidth. As corollaries we obtain the same result for any graph which satisfies the Ore-type condition d(x) + d(y) ≥ (1 + η)n for non-adjacent vertices x and y, or which satisfies a certain degree sequence condition. Thirdly, for γ > 0 we give a polynomial-time algorithm for determining whether or not a k-graph with minimum codegree at least (1/k + γ)n contains a perfect matching. This essentially answers a question of Rodl, Rucinski and Szemeredi. Our algorithm relies on a strengthening of a structural result of Keevash and Mycroft. Finally and additionally, we include a short note on Maker-Breaker games.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EPSRC)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics