The planarity of bipartite graphs in R

Abstract

Let R be a commutative ring and let Z(R) be its set of zero-divisors. We associate a graph Γ(R) to R with vertices Z(R)* = Z(R)-{0}, the set of non- zero zero divisors of R and for distinct u, v Z(R)*, the vertices u and v are adjacent if and only if uv = 0. In this paper, we evaluate the consistency of rectilinear crossing number of complete bipartite zero divisor graphs, in which the transformation of a non-planar graph into a planar graph is obtained by framing formula using removal of edges and removal of crossings.