Planar zero-divisor graphs

Abstract

This paper answers the question of Anderson, Frazier, Lauve, and Livingston: for which finite commutative rings R is the zero-divisor graph Γ (R) planar? We build upon and extend work of Akbari, Maimani, and Yassemi, who proved that if R is any local ring with more than 32 elements, and R is not a field, then Γ (R) is not planar. They left open the question: "Is it true that, for any local ring R of cardinality 32, which is not a field, Γ (R) is not planar?" In this paper we answer this question in the affirmative. We prove that if R is any local ring with more than 27 elements, and R is not a field, then Γ (R) is not planar. Moreover, we determine all finite commutative local rings whose zero-divisor graph is planar.

Local Rings with Genus Two Zero Divisor Graph

“…We will say that a graph G is a forbidden subgraph of k if G > k. We are concerned with finding the finite local commutative rings whose zero divisor graphs embed on 2 . The present work extends the efforts of [4,12], who classified the 0 embeddable rings, and [10,14], who did the same for 1 . In addition, [15] showed that if k is positive, then the set of finite commutative rings (up to isomorphism) whose zero divisor graphs have genus k is finite.…”

Finite commutative rings with higher genus unit graphs

“…The genus of graphs associated with rings is the topic of a number of publications. For instance, the planarity of zero divisor graphs were studied in [2,4,16]. The rings with toroidal zero divisor graphs were classified in Wang [19] and Wickham [21,22].…”

On the Genus of the Zero-Divisor Graph of Zn

“…Here our definition is the same as in [2], where some basic properties of Γ( ) are established. The zero-divisor graph, as well as other graphs of rings, is an active research topic in the last two decades (see, e.g., [3][4][5][6][7][8][9][10][11]). …”

“…Several papers focus on the genera of zero-divisor graphs. For instance, in [6,7,13,14], the authors studied the planar zerodivisor graphs (genus equals to 0); Wang et al investigated the genus one zero-divisor graphs in [11,15,16], respectively; and Bloomfield and Wickham determined all local rings whose zero-divisor graphs have genus two in [8]. In this paper, we study the zero-divisor graph of Z , the ring of integers modulo .…”