COMMUNICATIONS IN ALGEBRA
®
2017, VOL. 45, NO. 1, 275–284
http://dx.doi.org/10.1080/00927872.2016.1206344
Classification of nonlocal rings with genus one 3-zero-divisor
hypergraphs
K. Selvakumar and V. Ramanathan
Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India
ABSTRACT
Let R be a commutative ring with identity and let Z(R, k) be the set of all k-zero-
divisors in R and k > 2 an integer. The k-zero-divisor hypergraph of R, denoted
by H
k
(R), is a hypergraph with vertex set Z(R, k), and for distinct elements
x
1
, x
2
, ... , x
k
in Z(R, k), the set {x
1
, x
2
, ... , x
k
} is an edge of H
k
(R) if and only
if
k
i=1
x
i
= 0 and the product of any (k − 1) elements of {x
1
, x
2
, ... , x
k
} is
nonzero. In this paper, we characterize all finite commutative nonlocal rings R
with identity whose H
3
(R) has genus one.
ARTICLE HISTORY
Received 23 February 2015
Communicated by S. Sehgal
KEYWORDS
Hypergraph; incidence
graph; planar hypergraph;
toroidal graph; zero-divisor
graph
2010 MATHEMATICS
SUBJECT CLASSIFICATION
05C12; 05C25; 05C69; 13A99
1. Introduction
The study linking commutative ring theory with graph theory has been started with the concept of the
zero-divisor graph of a commutative ring. Let R be a commutative ring and Z(R)
∗
be the set of all nonzero
zero-divisors of R. The zero-divisor graph of R, denoted Ŵ(R), is the simple graph with Z(R)
∗
as the
vertex set and two distinct vertices x and y are joined by an edge if and only if xy = 0. This definition
was introduced by Beck, Anderson and Livingston in [5, 10] and later was studied extensively in
[1–6, 19, 23, 24]. In view of this, Eslahchi and Rahimi [15] have introduced and investigated a graph
called the k-zero-divisor hypergraph of a commutative ring. For a commutative ring R and k ≥ 2 a fixed
integer, a nonzero nonunit element x
1
in R is said to be a k-zero-divisor in R if there exist (k − 1) distinct
nonunit elements x
2
, x
3
, ... , x
k
in R different from x
1
such that
k
i=1
x
i
= 0 and the product of any
(k − 1) elements of {x
1
, x
2
, ... , x
k
} is nonzero. By Z(R, k), we denote the set of all k-zero-divisors of
R. The k-zero-divisor hypergraph H
k
(R) of R is defined as the hypergraph with the vertex set Z(R, k),
and for distinct elements x
1
, x
2
, ... , x
k
in Z(R, k), the set {x
1
, x
2
, ... , x
k
} is an edge of H
k
(R) if and only
if
k
i=1
x
i
= 0 and the product of any (k − 1) elements of {x
1
, x
2
, ... , x
k
} is nonzero. Note that the
graph constructed by 2-zero-divisors is exactly the same as the zero-divisor graph of a ring. It is shown
that for any finite nonlocal ring R, the hypergraph H
3
(R) is complete if and only if R is isomorphic to
Z
2
× Z
2
× Z
2
[15].
One of the most important topological properties of a graph is its genus. Finding the genus of a given
graph is a very hard problem, it is in fact NP-complete. The problem of finding the genus of a graph
associated with a ring has been studied by many authors; [1, 4, 8, 11, 13, 21, 24, 25, 28, 29], Tamizh
Chelvam et al. [26] determine all finite commutative nonlocal rings R for which H
3
(R) is planar. In this
paper, we characterize all finite commutative nonlocal rings R with identity whose H
3
(R) has genus one.
Throughout this paper, we assume that R is a finite commutative nonlocal ring with identity, Z(R),
its set of zero-divisors and R
×
, its group of units. We denote the ring of integers modulo n by Z
n
and the
field with q elements by F
q
. For any set X, let X
∗
denotes the nonzero elements of X. For basic definitions
on rings, one may refer [9, 18].
CONTACT K. Selvakumar selva_158@yahoo.co.in Department of Mathematics, Manonmaniam Sundaranar University,
Tirunelveli 627 012, Tamil Nadu, India.
© Manonmaniam Sundaranar University