Vertex and region colourings of planar idempotent divisor graphs of commutative rings

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, it is denoted by Π(R). The purpose of this work is to use some properties of ring theory and graph theory to determine the clique number, the chromatic number and the region chromatic number for each planar idempotent divisor graph of the commutative rings. furthermore, we show that the clique number is equal to the chromatic number for any planar idempotent divisor graph. Results indicate that when Fq and Fαp are fields of orders q and pα, respectively, where q=2 or 3, p is a prime number and is a positive integer. If ring R ∼= Fq ×Fpα, then χ(Π(R)) = ω(Π(R)) = χ∗(Π(R)) = 3. © 2022 Iraqi Journal for Computer Science and Mathematics. All rights reserved.