The chromatic numbers of prime graphs of polynomials and power series over rings

Walaa Alqarafi; Alaa Altassan; Wafaa Fakieh; Walaa Alqarafi; Alaa Altassan; Wafaa Fakieh

doi:10.3934/math.2025941

AIMS Mathematics

2025, Volume 10, Issue 9: 21061-21079. doi: 10.3934/math.2025941

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The chromatic numbers of prime graphs of polynomials and power series over rings

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received: 19 February 2025 Revised: 14 August 2025 Accepted: 03 September 2025 Published: 12 September 2025
MSC : 05C15, 05C25, 13F20, 13F25

A prime graph of a ring $ R $, denoted by $ PG^*(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(PG^*(R)) = \{ (x, y) : xRy = 0 $ or $ yRx = 0, x \neq y $ and $ x, y \in S(R) \} $. This graph is a subgraph of the prime graph $ PG(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 $ \chi(PG(R)) \leq n+1 $, where $ n $ is the order of the prime ideal. Moreover, we explore the chromatic number of the prime graph of $ M_2(\mathbb{Z}_n) $.
- prime graphs,
- polynomial rings,
- power series rings,
- chromatic numbers
Citation: Walaa Alqarafi, Alaa Altassan, Wafaa Fakieh. The chromatic numbers of prime graphs of polynomials and power series over rings[J]. AIMS Mathematics, 2025, 10(9): 21061-21079. doi: 10.3934/math.2025941

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Abstract

A prime graph of a ring $ R $, denoted by $ PG^*(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(PG^*(R)) = \{ (x, y) : xRy = 0 $ or $ yRx = 0, x \neq y $ and $ x, y \in S(R) \} $. This graph is a subgraph of the prime graph $ PG(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 $ \chi(PG(R)) \leq n+1 $, where $ n $ is the order of the prime ideal. Moreover, we explore the chromatic number of the prime graph of $ M_2(\mathbb{Z}_n) $.

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