Embedding spanning structures in graphs and hypergraphs

Knox, Fiachra (2013). Embedding spanning structures in graphs and hypergraphs. University of Birmingham. Ph.D.

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Abstract

In this thesis we prove three main results on embeddings of spanning subgraphs into graphs and hypergraphs. The first is that for log\(^5\)\(^0\) n/n \ ≤ p ≤ 1 – n \(^-\)\(^1\)\(^/\)\(^4\) log\(^9\) n, a binomial random graph G ~ G\(_n\)\(_,\)\(_p\) contains with high probability a collection of └\(\delta\)(G)/2┘ edge disjoint Hamilton cycles (plus an additional edge-disjoint matching if \(\delta\)(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 + \(\eta\))n for non-adjacent vertices x and y, or which satisfies a certain degree sequence condition. Thirdly, for \(\gamma\) > 0 we give a polynomial-time algorithm for determining whether or not a k-graph with minimum codegree at least (1/k + \(\gamma\))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.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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