K-Cycle Decomposition Of Complete Equal 7-Part Graph

Complete graph decomposition is an important research topic in graph theory.Let H1,H2,H3,…,HL be edge-disjoint subgraphs of G,if each subgraph is isomorphic to H and every edge of G exists exactly in a Hi,where 1≤i≤L,we call G decomposed into H,denoted as H|G.As one of the hot issues in graph decomposition,complete graph decomposition has attracted the attention of many scholars at home and abroad.They have done a lot of research on the decomposition of complete graph and have achieved a series of achievements.In this paper,we discuss the cycle decomposition of complete multipartite graphs.We mainly discuss the cycle decomposition of complete equal 7-part graphs(the size of each segment set is m)with edge disjoint length k(k≥3)if and only if k|21m2 and k≤7m.On the basis of E.J.Billington,B.R.Smith and N.J.Cavenagh’s study on the cycle decomposition of complete 5-part graphs,combined with the characteristics of complete equal 7-part graphs,the values of cycle length k which may be decomposed is classified into 8 cases,and the trace is obtained by proper decomposition of different subgraphs of complete equal 7-part graphs.Finally,these tracks are expanded by adding them to obtain conditional cycle decomposition.The main contents of this article are as follows:In the first part,the origin and development of graph theory and the research dynamic state of complete multipartite graph at home and abroad in recent years are introduced.In the second part,we mainly introduces the basic concept、symbol and theoretical foundation of graph theory.In the third part,According to the previous proof methods of cycle decomposition of complete 5-part graphs and the properties of complete equal 7-part graphs,we propose a necessary and sufficient condition for the existence of cycle decomposition of complete equal 7-part graph with length k,and give relevant proofs.