On Cycles in Directed Graphs

The main results of this thesis are the following. We show that for each α > 0 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least 3|G|/8 + α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen. Furthermore, answering completely a conjecture of Häggkvist 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 � ≥ 4 every sufficiently large oriented graph G with minimum indegree and minimum outdegree at least ≥ |G|/3+ 1 contains an �-cycle. This is best possible for all those � ≥ 4 which are not divisible by 3. Surprisingly, for some other values of �, an �-cycle is forced by a much weaker minimum degree condition. We propose and discuss a conjecture regarding the precise minimum degree which forces an �-cycle (with � ≥ 4 divisible by 3) in an oriented graph. We also give an application of our results to pancyclicity. This thesis is dedicated to Rachel. I can live with doubt and uncertainty and not knowing. I think it’s much more interesting to live not knowing than to have answers which might be wrong. I have approximate answers and possible beliefs and different degrees of certainty about different things, but I’m not absolutely sure of anything and there are many things I don’t know anything about, such as whether it means anything to ask why we’re here. I don’t have to know the answer. I don’t feel frightened by not knowing things, by being lost in a mysterious universe without any purpose, which is the way it really is as far as I can tell. It doesn’t frighten me. The Pleasure of Finding Things Out Richard Feynman, 1983