Kelly, Luke Tristian (2010)
Ph.D. thesis, University of Birmingham.
The main results of this thesis are the following. We show that for each alpha > 0 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least 3 |G| / 8 + alpha |G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. Furthermore, answering completely a conjecture of Haggkvist and Thomason, we show that we get every possible orientation of a Hamilton cycle. We also deal extensively with short cycles, showing that for each l > 4 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least |G| / 3 + 1 contains an l-cycle. This is best possible for all those l > 3 which are not divisible by 3. Surprisingly, for some other values of l, an l-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an l-cycle (with l > 3 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity.
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