THE 1-PLANARITY OF INTERSECTION GRAPH OF IDEALS OF A RING

Abstract

A graph G = (V (G), E(G)) is called 1-planar if it can be drawn in the plane such that every edge of the graph is cut by at most one other edge of the graph.  For any ring R, the ideal intersection graph of R, denoted by G(R), is the graph whose vertices are the nontrivial proper ideals of R and two distinct vertices are adjacent if they have nontrivial intersection.  In this paper, we characterize when the intersection graph G(R) of a ring R, is 1-planar.