Complete {3,4}-cycle Decompositions Of Directed Complete Graphs

In this thesis we extend Alspach conjecture from complete graphs Kv to direct-ed complete graphs Kv*, mainly research the necessary conditions of the conjecture are sufficient when mi∈{3,4}. At the present time, we completely resolve the existence of complete{3,4}—cycle decompositions of Kv*except when14≤v≤35. The main methods are as follows: when we prove the cases of v≡0,1,4,5,8,9(mod12), we use the methods of structuring a most useful directed graph (?), and decomposing directed complete graph into directed complete multigraphs and several directed complete graph with small cases, and combining with lemmas we proved in the thesis, after that we par-tition these multigraphs into (?), we get the conclusions at last. However, we resolve the remaining cases by using new methods of block design, mathematical induction and graph theory. For convenience of the explanation, we introduce the concept of admissible sequences and research its characters, then get some special skills, construction methods and some conclusions. Otherwise, there are some special cases to construct by myself. By using these methods and conclusions, we resolve the existence of complete {3,4}-cycle decompositions of Kv*.