Homogeneous Factorisations Of The Complete Graph Of Order A Product Of Two Distinct Primes

Before, experts studied cyclic homogeneous factorisations of complete graphs on the basis of self-complementary vertex transitive graphs.this paper further develops the isomorphic problem of cyclic homogeneous factorisations of com-plete graphs.to this, as the analysis of Caley graphs’ isomorphism, the paper define Cayley staining graph, and extends Adam’s conjecture, and comes to a conclusion, that for the complete graph of order a product of two distinct primes, Cay(Zpq,S,φ)= Cay(Zpq,S’,φ’),if and only if, there exists u ∈ Z*pq so that Si=uS’i (i=1,…,k). At last, the paper structs the cyclic homogeneous factorisations of Z13, Zg1, and achieves the count of non-isomorphism of homo-geneous factorisations. The full text is divided into 3 chapters, the main content is as follows.The chapter 1, introduct research background; and on the basis of defining Cayley staining graph, extend some basic concepts and properties, to prepare for the study of isormorphic problem.the chapter 2,determine necessary and sufficient condition of two cyclic ho-mogeneous factorisations’ isormorphism of complete graph of order prime, and achieve the counts of non-isormorphism of cyclic homogeneous factorisations of Z13.the chapter 3, determine necessary and sufficient condition of two cyclic ho-mogeneous factorisations’ isormorphism of complete graph of order a product of two distinct primes, and achieve the counts of non-isormorphism of cyclic homo-geneous factorisations of Z91.