Rings whose total graphs have small vertex-arboricity and arboricity

Year 2021, , 110 - 119, 04.02.2021

Morteza Fatehi Kazem Khashyarmanesh Abbas Mohammadian

https://doi.org/10.15672/hujms.589753

Abstract

Let $R$ be a commutative ring with non-zero identity, and $Z(R)$ be its set of all zero-divisors. The total graph of $R$, denoted by $T(\Gamma(R))$, is an undirected graph with all elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. In this article, we characterize, up to isomorphism, all of finite commutative rings whose total graphs have vertex-arboricity (arboricity) two or three. Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.

Keywords

Total graph‎, ‎arboricity‎, vertex-arboricity

References

• [1] D.F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320, 2706–2719, 2008.
• [2] D.F. Anderson and A. Badawi, The total graph of a commutative ring without the zero element, J. Algebra Appl. 11 1–18 pages, 2012.
• [3] G.J. Chang, C. Chen and Y. Chen, Vertex and tree arboricities of graphs, J. Comb. Optim. 8 295–306, 2004.
• [4] G. Chartrand, H.V. Kronk and C.E. Wall, The point arboricity of a graph, Israel J. Math. 6, 169–175, 1968.
• [5] T.T. Chelvam and T. Asir, On the genus of the total graph of a commutative ring, Comm. Algebra, 41, 142–153, 2013.
• [6] B. Corbas and G.D. Williams, Ring of order p5. II. Local rings, J. Algebra, 231 (2), 691–704, 2000.
• [7] H.R. Maimani, C. Wickham and S. Yassemi, Rings whose total graph have genus at most one, Rocky Mountain J. Math. 42, 1551–1560, 2012.
• [8] B.R. McDonald, Finite rings with identity , Pure Appl. Math. 28, Marcel Dekker, Inc., New York, 1974.
• [9] C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, Journal London Math. Soc, 39, 12, 1964.
• [10] R. Raghavendran, iFinite associative rings, Compositio Math. 21, 195–229, 1969.
• [11] S.P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math. 307, 1155–1166, 2007.