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The coloring of the cozero-divisor graph of a commutative ring

Akbari, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.1142/S1793830920500238
  3. Publisher: World Scientific , 2020
  4. Abstract:
  5. Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*-(R), where W*-(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a*‰Rb and b*‰Ra. Let ω(Γ′(R)) and χ(Γ′(R)) denote the clique number and the chromatic number of Γ′(R), respectively. In this paper, we prove that if R is a finite commutative ring, then Γ′(R) is perfect. Also, we prove that if R is a commutative Artinian non-local ring and ω(Γ′(R)) is finite, then χ(Γ′(R)) = ω(Γ′(R)). For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if R is a commutative ring, then Γ′(R) is a complete bipartite graph if and only if R≠F1 × F2, where F1 and F2 are fields. Moreover, we present some results on the complete r-partite cozero-divisor graphs. © 2020 World Scientific Publishing Company
  6. Keywords:
  7. Chromatic number ; Clique number ; Cozero-divisor graph
  8. Source: Discrete Mathematics, Algorithms and Applications ; Volume 12, Issue 3 , 2020
  9. URL: https://www.worldscientific.com/doi/abs/10.1142/S1793830920500238