On The Crossing Numbers Of Some Graphs

The crossing number of graphs is a vital concept in modern graph theory and a characterization of a graph of non-plane of an important parameter. It is difficulty in the forefront, of topological graph theory. The crossing number of graphs comes from the Pual Turdn’s brick factory problem during in the fifties of the J9th century, and it becomes a very active branch in Graph Theory, whose theory has widespread applications in many fields, such as the design of electronic circuits, and the identification and repaint of sketch, and the graphical representation of DMA in biology engineer and so on. Many authors at home and aborad have engaged in the investigation on this area. It mainly studied how to take a. graph in the plane or curved surface, to make its crossing number at least, usually used the pure mathematical method to proof it. Generally, it is difficultly to determine the crossing number of graph. In fact. Garev and Johnson have proved that in general the problem of determining the crossing number of a graphs is NP-complete.At present. the classes of graphs whose crossing number have been determined are very scarce, and there are only some special graphs whose crossing numbers are known. For example, these included the complete graph Kn. the complete bipartite graph Km,n. the complete tripartite graph Kl.m.n, Some Cartesian product between some simply graphs, certain Petersen graph and so on. In this paper, we study the crossing number of some special graphs, such as the generalized Petersen graph, the join product, the Cartesian product, etc,with some new methods. The details are as follows:One. We introduce the origins and backgrounds of the crossing number. its theorical and practical meanings, its development around the world.Two. Introduce some basic concepts as well as some lemma and proper-ties,which associated with the graph.Three. Focusing on the crossing number of join product, determines that the crossing number of the join product of a typical six vertices graphs with two hangs edges with Path and Cycle.Four. Proof the crossing number of the join product of a special graphs with six vertices with Path and Cycle.Five. Studies the crossing number of the Wheel, determines the crossing numbers of wheel-W3join with path Pn and Cartesian products of wheel-W6with Sn.provided that Zarankiewiez’s conjecture holds for the ease m-7.Six. Studies the crossing number of a type of Petensen graph. determined the upper and lower bounds of the generalized Petersen graph P (3k1k) for arbitrary k>3.Seven. We make some conclusions of our research work and put forward one relative problems which we will go ahead.