@article{Alqarafi2025, 
author = {Walaa Alqarafi and Alaa Altassan and Wafaa Fakieh},
title = {The chromatic numbers of prime graphs of polynomials and power series over rings},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {9},
pages = {21061-21079},
keywords = {prime graphs, polynomial rings, power series rings, chromatic numbers},
url = {https://www.sciopen.com/article/10.3934/math.2025941},
doi = {10.3934/math.2025941},
abstract = {A prime graph of a ring    R, denoted by    P      G    ∗    (  R  ), is a graph whose vertex set is the set of the strong zero divisors    S  (  R  ) of    R, and its edge set is either    E  (  P      G    ∗    (  R  )  )  =  {  (  x  ,  y  )  :  x  R  y  =  0 or    y  R  x  =  0  ,  x  ≠  y and    x  ,  y  ∈  S  (  R  )  }. This graph is a subgraph of the prime graph    P  G  (  R  ). In this paper, we investigate the chromatic numbers of the prime graphs of Artinian rings that satisfy certain conditions. In particular, if    R is an Artinian ring with a unique prime ideal, then we prove that    χ  (  P  G  (  R  )  )  ≤  n  +  1, where    n is the order of the prime ideal. Moreover, we explore the chromatic number of the prime graph of        M    2    (            Z        n    ).}
}