Abstract
The idempotent divisor graph of a commutative ring R is a graph with vertices set in R * = R-{0}, andany distinct vertices x and y are adjacent if and only if x.y = e. For some non-unit idempotent elemente2=e∈R, it isdenoted byΠ(R) . The purpose of this work is to use some properties of ring theory and graph theory to determine theclique number, the chromatic number and the region chromatic number for each planar idempotent divisor graph ofthe commutative rings. furthermore, we show that the clique number is equal to the chromatic number for any planaridempotent divisor graph. Results indicate that when Fqand Fαpare fields of ordersqandpα, respectively, where q=2or 3, p is a prime number and is a positive integer. If ringR∼=Fq×Fαp,thenχ(Π(R)) =ω(Π(R)) =χ∗(Π(R)) =3
Recommended Citation
Authman, Mohammed N.; Mohammad, Husam Q.; and Shuker, Nazar H.
(2022)
"Vertex and region colourings of planar idempotent divisor graphsof commutative rings.,"
Iraqi Journal for Computer Science and Mathematics: Vol. 3:
Iss.
1, Article 8.
DOI: https://doi.org/10.52866/ijcsm.2022.01.01.008
Available at:
https://ijcsm.researchcommons.org/ijcsm/vol3/iss1/8
