Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

Abstract

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ? R, and is denoted by ?(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (?(R))= (?(R)) = *( ?(R)) = 3.

Keywords

• Zero divisor graph, idempotent elements, planar graph and idempotent divisor graph.