Planar zero-divisor graphs

By Richard G. Belshoff, J. Chapman. 2007

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Commutative rings with toroidal zero-divisor graphs (2007, background): “. Therefore Γ(R) = ∅ if and only if Ris an integral domain. This deﬁnition was introduced by Anderson and Livingston in [6]. Recently, this subject has been extensively studied in [2], [3], [4], [5], [7], [8], [12], [13], [15], [18], [19], and [20]. There are many known results concerning zero-divis”
On Zero Divisor Graphs (2012, background): “Smith [33], independently provided a complete answer, as well as a classification of which rings are planar, listing forty-four isomorphism classes in all.”
On the Zero-Divisor Graph of a Ring (2008, background): “For other articles on zero-divisor graphs, see Anderson and Naseer (1993), Anderson (2008), Anderson et al. (2001, 2003), Anderson and Mulay (2007), Axtel et al. (2005), Axtel and Stickles (2006), DeMeyer and Schneider (2002), Lucas (2006), Mulay (2002), Redmond (2007), and Smith (2003).”
Classification of Rings with Genus One Zero-Divisor Graphs (2008, background): “In [3], the question was asked: For which flnite commutative rings R is i(R) planar? A partial answer was given in [1], but the question remained open for local rings of order 32. In [20], and independently in both [21] and [ 6 ], it was shown that no local ring of order 32 has a planar zero-divisor”
On the Genus of the Total Graph of a Commutative Ring (2013, background): “The planarity of zero divisor graphs were studied in [2, 3, 8, 16].”