The Fixed-length Cycle Decomposition Of Complete Equal-6 Graphs

The development and research of graph theory have great convenience to our real life.The goal of graph theory is to present abstract problems as simple fundamental problems.The object of graph theory is graph.According to the research goal,it is to show the abstract problem in the form of graph.Cycle decomposition is one of the most important problems in graph theory.Cycle decomposition of complete isomeric graphs,2009,Elizabeth J.Billington,Nicholas J.Cavenagh and Benjamin R.In 2010,it was also proved that complete equal 3-part graphs and complete equal 5-part graphs have Necessity and sufficiency of cycle decomposition.In this paper,based on the cycle decomposition of complete isomorphic 4-partite graphs,we give the existence and necessity,sufficiency of the fixed length cycle de-composition of complete 6-partite graphs.The existence of fixed length k cycle de-composition of a complete six-partite graph is k|60m2 and k?12m.In this paper,according to the definition of fixed-length cycle decomposition and the relationship between the number of edges and the degree of vertices in the structure of a complete isomorphic 6-partite graph,we find out the length k where there might be a k cycle decomposition,in order to prove the existence of this cycle,kmust satisfy k|60m2 and k?12m.The key of the proof is to construct the closed trace of the corresponding length according to the length of different kcycles,and then obtain the proof of the closed trace of the corresponding cycle decomposition by the relevant Lemma.The following is the main structure and research content of this paper.In the first chapter,we mainly introduce the related fields of this thesis,elaborate the research background and present situation of cycle decomposition of graphs,and briefly describe the main results of this thesis.In the second chapter,we give some basic concepts,related theorems and lemmas needed in this paper.In the third chapter,we mainly study the structure of complete graphs K(2m,2m,2m,2m,2m,2m),and give six main theorems of cycle decomposition of complete iso-morphic graphs with even number of partial sets.