The Uniqueness Of Skeleton Presentation Of Complete Bipartite Graph K_m,n

In1983, J H. Conway and C.McA. Gordon showed that every embedding of the complete graph K7in the three-dimensional Euclidean space R3contains a knotted cycle. And Miki Shimabara has proved that every embedding of the complete bipartite graph K5,5in R3contains a knotted cycle. In the second part of this paper we generalized their method and show that every embedding of the complete tripartite graph K3,3,3in R3contains a knotted cycle.Kobayashi discussed some kinds of standard embeddings into3-manifolds of spatial graphs. He introduced the concept of book presentation, which is a standard embedding of spatial graphs with good properties, and proved that the book presentation of mini-mum sheets of Kn is unique up to the sheet translation and the ambient isotopy. Yin Xunbo,Long Ligang,Lei Fengchun shown some properties of book presentation of spatial graphs, and prove that the book presentation of minimum sheets of a complete graph K2m with even vertices is unique up to sheet translation and ambient isotopy.In the third part of this present paper we give the definition of skeleton presenta-tion of spatial graphs, and prove that the skeleton presentation of minimum floors of a complete bipartite graph Km,n is unique up to ambient isotopy.G.S.Bloom, J.W.Kennedy and L.V.Quintas shown that if G is a cubic graph that does not embed in the projective plane, then G contains one of the six graphs. H.H.Glover and J.P.Huneke proved that every cubic graph that embed in the projective plane with crossing number at least two contains a subdivision of one of two graphs.In the forth part of this present paper we proved that every cubic graph that embed in the projective plane with crossing number at least four contains a subdivision of one of two graphs with crossing number four.