Cycle partitions and path decompositions in directed graphs

This thesis makes progress on three conjectures in extremal graph theory: one concerning the Hamiltonicity of regular oriented graphs, another on the Hamiltonicity of regular directed graphs, and a third about path decompositions of oriented graphs. A Hamilton cycle in a (directed) graph is a (directed) cycle that visits every vertex of the (directed) graph.
The first result concerns a conjecture of Jackson from 1981 which asserts that every d-regular oriented graph on n vertices with n<=4d+1 has a Hamilton cycle. We confirm this conjecture for large n by proving a more general result on cycle partitions in dense regular directed and oriented graphs.
The second result addresses a conjecture of Kühn and Osthus from 2010 which states that every strongly 2-connected d-regular directed graph on n vertices with d>=n/3 has a Hamilton cycle. We disprove this conjecture by providing a counterexample, but prove that a slightly modified version holds approximately.
Lastly, we consider a conjecture from 1980 attributed to Pullman on path decompositions of orientations of regular graphs. There is a natural lower bound on the number of paths needed in any path decomposition of a directed graph, based on its degree sequence. The conjecture states that for every fixed odd d, this bound is achieved for every orientation of every d-regular graph. We prove that the conjecture holds for random regular graphs with high probability; that is for fixed odd d, as n tends to infinity, it holds for almost all d-regular graphs on n vertices.