Zero-Divisor Graph of an Ideal of a Near-Ring

In this paper, we associate the graph ΓI(N) to an ideal I of a near-ring N. We exhibit some properties and structure of ΓI(N). For a commutative ring R, Beck conjectured that both chromatic number and clique number of the zero-divisor graph Γ(R) of R are equal. We prove that Beck's conjecture is true for ΓI(N). Moreover, we characterize all right permutable near-rings N for which the graph ΓI(N) is finitely colorable.

[1]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[2]  S. Veldsman.  On equiprime near-rings , 1992 .

[3]  Saieed Akbari,et al.  The total graph and regular graph of a commutative ring , 2009 .

[4]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[5]  David F. Anderson,et al.  The Zero-Divisor Graph of a Commutative Ring☆ , 1999 .

[6]  G. Birkenmeier,et al.  Completely Prime Ideals and Radicals in Near-Rings , 1995 .

[7]  Shane P. Redmond,et al.  An Ideal-Based Zero-Divisor Graph of a Commutative Ring , 2003 .

[8]  P. Dheena,et al.  AN IDEAL-BASED ZERO-DIVISOR GRAPH OF 2-PRIMAL NEAR-RINGS , 2009 .

[9]  Ayman Badawi,et al.  The total graph of a commutative ring , 2008 .

[10]  H. Heatherly,et al.  Medial near-rings , 1989 .

[11]  D. D. Anderson,et al.  Beck′s Coloring of a Commutative Ring , 1993 .

[12]  Hamid Reza Maimani,et al.  Zero-Divisor Graph with Respect to an Ideal , 2006 .

[13]  I. Beck.  Coloring of commutative rings , 1988 .

[14]  B. S. Kedukodi,et al.  Graph of a Nearring with Respect to an Ideal , 2010 .

[15]  T. T. Chelvam,et al.  On the Genus of the Total Graph of a Commutative Ring , 2013 .

[16]  Karim Samei,et al.  The zero-divisor graph of a reduced ring , 2007 .

[17]  T. Tamizh Chelvam,et al.  Intersection graph of gamma sets in the total graph , 2012, Discuss. Math. Graph Theory.

[18]  Shane P. Redmond,et al.  Zero-Divisor Graphs of Nearrings and Semigroups , 2005 .

[19]  David F. Anderson,et al.  Zero-divisor graphs, von Neumann regular rings, and Boolean algebras , 2003 .