The Problem Of Embedding On The Extended Grid Of The Torus Knots T_3,2N And T（4,2N） And The Crossing Number Of K（2,5N）

Valiant showed that a graph G embeds in a grid if and only if it is planar of maximum degree at most four.By splitting each vertex of a2-connected planar graph G, then the degree of each vertex of G is at most3. A graph is Y△Y-reducible if it can be reduced to a vertex by a sequence of series-parallel reductions and Y△Y-tranformations. But they need the minimum number of steps in this progress have not been clear. In this paper we summarize a regular that the standard projective graph of tours knot T3,2n and the tours knot T4,2n can be embed into a extended grid and that the minimum number of steps when it can be reduced to a vertex.Zarankiewicz asserted that the crossing number of the complete bipartite graph Km,n (m≤n) is [m/2][(m-1)/2][n/2][(n-1)/2], which is known only for m≤6, or m=7and≤10, if Zarankiewicz’s conjecture is correct for m=7and n which is arbitrary. In this paper, we prove the crossing number of K2,5,n.