Elsevier

Journal of Algebra

Volume 274, Issue 2, 15 April 2004, Pages 847-855
Journal of Algebra

On the zero-divisor graph of a commutative ring

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Abstract

Let R be a commutative ring and Γ(R) be its zero-divisor graph. In this paper it is shown that for any finite commutative ring R, the edge chromatic number of Γ(R) is equal to the maximum degree of Γ(R), unless Γ(R) is a complete graph of odd order. In [D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, in: Lecture Notes in Pure and Appl. Math., Vol. 220, Marcel Dekker, New York, 2001, pp. 61–72] it has been proved that if R and S are finite reduced rings which are not fields, then Γ(R)≃Γ(S) if and only if RS. Here we generalize this result and prove that if R is a finite reduced ring which is not isomorphic to Z2×Z2 or to Z6 and S is a ring such that Γ(R)≃Γ(S), then RS.

Keywords

Zero-divisor graph
Edge coloring
Hamiltonian

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