Table Of Content. .
RINGS WHOSE TOTAL GRAPHS HAVE GENUS AT MOST ONE
HAMIDREZAMAIMANI,CAMERONWICKHAM,ANDSIAMAKYASSEMI
0
1 Abstract. Let R be a commutative ring with Z(R) its set of zero-divisors.
0 In this paper, we study the total graph of R, denoted by T(Γ(R)). It is
2 the (undirected) graph with all elements of R as vertices, and for distinct
x,y ∈ R, the vertices x and y are adjacent if and only if x+y ∈ Z(R). We
n
investigate properties of the total graph of R and determine all isomorphism
a
classes of finite commutative rings whose total graph has genus at most one
J
(i.e.,aplanarortoroidalgraph). Inaddition,itisshownthat,givenapositive
9 integerg,thereareonlyfinitelymanyfiniteringswhosetotalgraphhasgenus
2 g.
]
C
A
. Introduction
h
t
a Let R be a commutative ring with non-zero unity. Let Z(R) be the set of zero-
m divisorsofR. TheconceptofthegraphofthezerodivisorsofRwasfirstintroduced
byBeck[6],wherehewasmainlyinterestedincolorings. Inhisworkallelementsof
[
theringwereverticesofthegraph. Thisinvestigationofcoloringsofacommutative
1
ringwasthencontinuedbyD.D.AndersonandNaseerin[5]. In[4],D.F.Anderson
v
and Livingston associate a graph, Γ(R), to R with vertices Z(R)\{0}, the set of
8
3 non-zero zero-divisorsof R, and for distinct x,y ∈Z(R)\{0},the vertices x and y
3 are adjacent if and only if xy =0.
5 An interesting question was proposed by Anderson, Frazier, Lauve, and Liv-
.
1 ingston[3]: For whichfinite commutativerings R is Γ(R) planar. Apartialanswer
0 was givenin [1], but the question remained open for local rings of order 32. In [12]
0 and then independently in [7] and [13] it is shown that there is no ring of order 32
1
whose zero-divisor graph is planar.
:
v The genusofagraphisthe minimalintegern suchthatthe graphcanbe drawn
i
X withoutcrossingitself ona spherewith n handles(i.e. anorientedsurfaceofgenus
n). Thus, a planar graph has genus zero, because it can be drawn on a sphere
r
a without self-crossing. In [13] and [16] the rings whose zero-divisorgraphhas genus
one are studied. A genus one graph is called a toroidal graph. In other words, a
graph G is toroidal if it can be embedded on the torus, that means, the graph’s
vertices can be placed on a torus such that no edges cross. Usually, it is assumed
thatGis alsonon-planar. In[17] itis shownthatfor apositive integerg, thereare
only finitely many finite rings whose zero-divisor graph has genus g.
In [2], D. F. Anderson and Badawi introduced the total graph of R, denoted by
T(Γ(R)), as the graph with all elements of R as vertices, and for distinct x,y ∈R,
the vertices x and y are adjacent if and only if x+y ∈Z(R).
2000 Mathematics Subject Classification. 05C75,13A15.
Key words and phrases. Totalgraph,genus, planargraph,toroidalgraph.
TheresearchofHamidMaimaniwasinpartsupportedbyagrantfromIPM(No. 88050214).
TheresearchofSiamakYassemiwasinpartsupportedbyagrantfromIPM(No. 88130213).
1
2 HAMIDREZAMAIMANI,CAMERONWICKHAM,ANDSIAMAKYASSEMI
Inthispaper,weinvestigatepropertiesofthetotalgraphofRanddetermineall
isomorphismclasses of finite rings whose total graphhas genus at most one (i.e., a
planar or toroidal graph). In addition, we show that for a positive integer g, there
are only finitely many finite rings whose total graph has genus g.
1. Main result
A complete graph is a graph in which each pair of distinct vertices is joined
by an edge. We denote the complete graph with n vertices by K . A bipartite
n
graph is a graph such that its vertex set can be partitioned into two subsets V
1
and V and each edge joins a vertex of V to a vertex of V . A complete bipartite
2 1 2
graph is a bipartite graph such that each vertex in V is joined by an edge to
1
each vertex in V , and is denoted by K when |V | = m and |V | = n. A
2 m,n 1 2
clique of a graph is a maximal complete subgraph. For a graph G, the degree of a
vertex v in G, denoted deg(v), is the number of edges of G incident with v. The
numberδ(G)=min{deg(v)|v is a vertex of G}istheminimum degree ofG. Fora
nonnegativeintegerk,agraphiscalledk-regularifeveryvertexhasdegreek. Recall
that a graph is said to be connected if for each pair of distinct vertices v and w,
thereisafinitesequenceofdistinctverticesv =v ,··· ,v =w suchthateachpair
1 n
{v ,v } is an edge. Such a sequence is said to be a path and the distance d(v,w)
i i+1
between connected vertices v and w is the length of the shortest path connecting
them. For any graph G, the disjoint union of k copies of G is denoted kG. Let S
be a nonempty subset of vertex set of graph G. The subgraph induced by S is the
subgraph with the vertex set S and with any edges whose endpoints are both in
the S and is denoted by hSi.
Let G =(V ,E ) and G =(V ,E ) be two graphs with disjoint vertices set V
1 1 1 2 2 2 i
and edges set E . The cartesianproduct of G and G is denoted by G=G ×G
i 1 2 1 2
withverticessetV ×V and(x,y) is adjacentto (x′,y′)if x=x′ andy is adjacent
1 2
y′ in G or y =y′ and x is adjacent to x′ in G .
2 1
Lemma 1.1. Let x be a vertex of T(Γ(R)). Then the degree of x is either |Z(R)|
or |Z(R)|−1. In particular, if 2 ∈ Z(R), then T(Γ(R)) is a (|Z(R)|−1)-regular
graph.
Proof. If x is adjacent to y, then x+y = a ∈ Z(R) and hence y = a−x for some
a∈Z(R). We have two cases:
Case 1. Suppose 2x∈Z(R). Then x is adjacent to a−x for any a∈Z(R)\{2x}.
Thus the degree of x is |Z(R)|−1. In particular, if 2 ∈ Z(R), then T(Γ(R)) is a
(|Z(R)|−1)-regular graph.
Case 2. Suppose 2x ∈/ Z(R). Then x is adjacent to a−x for any a ∈ Z(R). Thus
the degree of x is |Z(R)|. (cid:3)
LetS denotethe spherewith k handles,wherek is anon-negativeinteger,that
k
is, S is an oriented surface of genus k. The genus of a graph G, denoted by γ(G),
k
is the minimum integer n such that the graph can embedded in S . A graph G is
n
called a planar if γ(Γ(G)) = 0, and toroidal if γ(Γ(G)) = 1. We note here that if
H is a subgraph of a graph G, then γ(H)≤γ(G).
In the following theorem we bring some well-known formulas, see, e.g., [14] and
[15]:
Theorem 1.2. The following statements hold:
RINGS WHOSE TOTAL GRAPHS HAVE GENUS AT MOST ONE 3
(a) For n≥3 we have γ(K )= (n−3)(n−4) .
n 12
l m
(b) For m,n≥2 we have γ(K )= (m−2)(n−2) .
m,n 4
l m
(c) Let G and G be two graphs and for each i, p be the number of vertices
1 2 i
of G . Then max{p γ(G )+γ(G ),p γ(G )+γ(G )}≤γ(G ×G ).
i 1 2 1 2 1 2 1 2
According to Theorem 1.2 we have γ(K )=0 for 1≤n≤4 and γ(K )=1 for
n n
5≤n≤7 and for other value of n, γ(K )≥2.
n
Lemma 1.3. Let F denote the field with q elements. Then the total graph of
q
F ×F is isomorphic to K ×K . Furthermore, for any positive integer m and
2 q 2 q
q >2,
(q−3)(q−4)
γ(T(Γ(F2m ×Fq)))≥2m .
(cid:24) 12 (cid:25)
Proof. We induct on m. If m=1, let V ={(0,y)|y ∈F } and let V ={(1,y)|y ∈
0 q 1
F }. Then the subgraphs G of the total graph of F × F induced by each of
q i 2 q
the V is K . Now for each y ∈ F , there is an edge between (0,y) ∈ G and
i q q 0
(1,−y) ∈ G . Furthermore, these are the only other edges in the total graph.
1
Identifying (1,−y) with (1,y), we can replace G with an isomorphic copy G′;
1 1
under this isomorphism the edge between (0,y)∈G and (1,−y)∈G is the edge
0 1
between (0,y) ∈ G and (1,y) ∈ G′. Thus the total graph of F ×F has vertex
0 1 2 q
set {(x,y)|x ∈ F and y ∈ F }, with an edge between (x,y) and (x′,y′) if x = x′
2 q
and y 6= y′, or y =y′ and x6=x′. That is, it is the graph K ×K . Parts (a) and
2 q
(c) of Theorem 1.2 now yield
(q−3)(q−4)
γ(T(Γ(F ×F )))=γ(K ×K )≥2 .
2 q 2 q
(cid:24) 12 (cid:25)
If m>1, we can partition F2m into two sets, S1 and S2, each of cardinality 2m−1;
let f be a bijection from S to S . Since each element of a field of characteristic
1 2
2 is its own inverse, then the subgraph of T(Γ(F2m ×Fq)) induced by Si ×Fq is
isomorphic to T(Γ(F2m−1 ×Fq)). For any y ∈ Fq and s ∈ S1, the element (s,y)
is adjacent to (f(s),−y). We thus have a copy of K2 ×T(Γ(F2m−1 ×Fq)) as a
subgraphofT(Γ(F2m×Fq)). Part(c)ofTheorem1.2andtheinductionhypothesis
now yield
γ(T(Γ(F2m ×Fq))) ≥ γ(K2×T(Γ(F2m−1 ×Fq)))
≥ 2γ(T(Γ(F2m−1 ×Fq)))
(q−3)(q−4)
≥ 2m .
(cid:24) 12 (cid:25)
(cid:3)
A subdivision of a graph is a graph obtained from it by replacing edges with
pairwise internally disjoint paths. A remarkably simple characterization of planar
graphs was given by Kuratowski in 1930. Kuratowski Theorem says that a graph
is planar if and only if it contains no subdivision of K or K (see [2, p. 153]). In
5 3,3
addition, every planar graph has a vertex v such that deg(v)≤5.
Theorem 1.4. For any positive integer g, There are finitely many finite rings R
whose total graph has genus g.
4 HAMIDREZAMAIMANI,CAMERONWICKHAM,ANDSIAMAKYASSEMI
Proof. Let R be a finite ring. If R is local, then Z(R) is the maximal ideal of
R and |R| ≤ |Z(R)|2. If R is not local, then R = R × R × ···× R where
1 2 n
each of the R ’s is a local ring and n ≥ 2 [10]. Suppose that |R | ≤ |R | ≤
i 1 2
··· ≤ |R | and set R∗ = 0×R ×···R . Since |R| = |R ||R∗|, we conclude that
n 1 2 n 1 1
|R| ≤ |R∗|2. Let S denote either Z(R) if R is local or R∗ if R is not local. Then
1 1
every pair of elements of S are adjacent in T(Γ(R)) and hence we have a complete
graph K in the structure of T(Γ(R)). This implies that γ(K ) ≤ g. Therefore
|S| |S|
⌈((|S|−3)(|S|−4))/12⌉ ≤ g and so |S| ≤ 7+ 49+48(g−1) /2, and hence
2 (cid:16) p (cid:17)
|R|≤ 7+ 49+48(g−1) /2 . (cid:3)
(cid:16)(cid:16) p (cid:17) (cid:17)
Theorem 1.5. Let R be a finite ring such that T(Γ(R)) is planar. Then the
following hold:
(a) If R is local ring, then R is a field or R is isomorphic to the one of the 9
following rings:
Z ,Z [X]/(X2),Z [X]/(X3),Z [X,Y]/(X,Y)2,Z [X]/(2X,X2),
4 2 2 2 4
Z [X]/(2X,X2−2),Z ,F [X](X2),Z [X]/(X2+X +1).
4 8 4 4
(b) If R is not local ring, then R is an infinite integral domain or R is isomor-
phic to Z ×Z or Z .
2 2 6
Proof. Anyplanargraphhasavertexv withdeg(v)≤5. So ifthe totalgraphofR
isplanar,thenδ(T(Γ(R)))≤5. ByLemma1.1,δ(T(Γ(R)))=|Z(R)|or|Z(R)|−1,
and hence |Z(R)|≤6.
(a) Assume that R is a local ring and let n = |Z(R)| and m = |R/Z(R)|. If
2 ∈ Z(R) then T(Γ(R)) ∼= mKn ([2, Theorems 2.1 and 2.2]). Hence |Z(R)| ≤ 4.
Also, |R| = 2k since 2 ∈ Z(R). So |R| = 16,8,4, or 2. According to Corbas and
Williams [8] there are two non-isomorphic rings of order 16 with maximal ideals
of order 4, namely F [x]/(x2) and Z [x]/(x2+x+1) (see also Redmond [11]), so
4 4
for these rings we have T(Γ(R)) ∼= 4K . Since K is planar we conclude that the
4 4
total graphs of these rings are planar. In [8] it is also shown that there are 5 local
rings of order 8 (except F ). In all of these rings we have |Z(R)| = 4 and hence
8
T(Γ(R))∼=2K . Also,therearetwonon-isomorphiclocalringsoforder4;theseare
4
Z and Z [X]/(X2). For both we have T(Γ(R)) ∼= 2K and thus they are planar.
4 2 2
Note that if |Z(R)| = 1, then R is a field and hence the total graph is planar.
If 2 ∈/ Z(R), then T(Γ(R)) ∼= Kn∪ m2−1 Kn,n ([2, Theorem 2.2]). This implies
n≤2, and thus R either has order 4(cid:0)or is(cid:1)a field.
(b) Suppose that R is not local ring. Since R is finite, then there are finite local
rings R such that R=R ×···×R where t≥2. Since |Z(R)|≤6 then we have
i 1 t
the following candidates:
Z ×Z ,Z ,Z ×F ,Z ×Z ,Z ×Z [X]/(X2),
2 2 6 2 4 2 4 2 2
Z ×Z ,Z ×Z ,Z ×F .
2 5 3 3 3 4
The total graph of Z ×Z is isomorphic to the cycle C , and this graphis planar.
2 2 4
By Lemma 1.3, the total graph of Z ∼= Z ×Z is isomorphic to K ×K , which
6 2 3 2 3
is also planar.
LetRbearingwith|R|=n. ThesubgraphofthetotalgraphofZ ×Rinduced
2
bytheset{0}×RisacopyofK . Theedge(1,0)−(0,0),togetherwiththe paths
n
(1,0)−(1,−r)− (0,r) for each r ∈ R yield a subdivision of K in the total
n+1
RINGS WHOSE TOTAL GRAPHS HAVE GENUS AT MOST ONE 5
graphofZ ×R. Thus the totalgraphsof Z ×F , Z ×Z , Z ×Z [X]/(X2), and
2 2 4 2 4 2 2
Z ×Z are not planar. Also, the total graphof Z ×R contains a subgraphwhich
2 5 3
is isomorphic to K (consider the induced subgraph hSi where S = {(1,r)|r ∈
3,n
R}∪ {(2,r)| r ∈ R}). Thus the total graphs of Z × Z and Z × F are not
3 3 3 4
planar. (cid:3)
Theorem 1.6. Let R be a finite ring such that T(Γ(R)) is toroidal. Then the
following statements hold:
(a) If R is local ring, then R is isomorphic to Z , or Z [X]/(X2).
9 3
(b) If R is not local ring, then R is isomorphic to one of the following rings:
Z ×F ,Z ×Z ,Z ×Z ,Z ×Z [x]/(x2),Z ×Z ×Z
2 4 3 3 2 4 2 2 2 2 2
Proof. ForanygraphGwithν verticesandgenusgwehaveδ(G)≤6+(12g−12)/ν.
If γ(G)=1, then δ(G)≤6 and equality holds if and only if G is a triangulation of
thetorusand6-regular(see[16,Proposition2.1]). IfRisafiniteringwithtoroidal
total graph, then δ(T(Γ(R))) ≤ 6, and by Lemma 1.1 δ(T(Γ(R))) = |Z(R)| or
|Z(R)|−1. Thus we conclude that |Z(G)|≤7.
(a) Let R be a local ring. If 2 ∈ Z(R), then T(Γ(R)) is a disjoint union of
copies of the complete graph K , where |Z(R)| = n. Hence 5 ≤ n ≤ 7. But
n
in this case |Z(R)| is a power of 2 and thus there are no such local rings. Now
suppose that 2∈/ Z(R). Then T(Γ(R))∼=Kn∪ m2−1 Kn,n, where n=|Z(R)| and
m = |R/Z(R)|. Thus 3 ≤ n ≤ 4 and since 2 ∈/(cid:0)Z(R)(cid:1)we must have n = 3. There
are two local rings, Z and Z [X]/(X2), such that the cardinality of the set of the
9 3
zero-divisorsis 3; both of these rings have total graphK ∪K which is toroidal.
3 3,3
(b) Assume that R is not a local ring. Since |Z(R)| ≤ 7, we have the following
candidates for R by Theorem 1.5(b):
Z ×F ,Z ×Z ,Z ×Z [x]/(x2),Z ×Z ,Z ×Z ,Z ×F ,F ×F ,Z ×Z ×Z .
2 4 2 4 2 2 2 5 3 3 3 4 4 4 2 2 2
By Theorem 1.5, γ(T(Γ(Z ×F ))),γ(T(Γ(Z ×Z ))),γ(T(Γ(Z ×Z ))) are all at
2 4 2 4 3 3
least 1. The embeddings in Figure 1 parts (a), (b), and (c) show explicitly that
γ(T(Γ(Z ×F ))) = γ(T(Γ(Z ×Z ))) = γ(T(Γ(Z ×Z ))) = 1. Since T(Γ(Z ×
2 4 2 4 3 3 2
Z [x]/(x2)))∼=T(Γ(Z ×Z )), then γ(T(Γ(Z ×Z [x]/(x2))))=1.
2 2 4 2 2
IfwepartitiontheelementsofZ ×Z ×Z bythefoursetsV ={(0,0,0),(1,1,1)},
2 2 2 1
V = {(1,0,0),(0,1,1)}, V = {(0,1,0),(1,0,1)}, and V = {(0,0,1),(1,1,0)}, it
2 3 4
is clear that T(Γ(Z ×Z ×Z )) = K . Hence γ(Z ×Z ×Z ) = 1 by [9,
2 2 2 2,2,2,2 2 2 2
Corollary 4]
By Lemma 1.3, γ(T(Γ(Z ×Z ))) ≥ 2 and thus is not toroidal. By (the proof
2 5
of) Lemma 1.3, γ(T(Γ(F ×F ))) ≥ 2γ(T(Γ(Z ×F ))) and so by Theorem 1.5
4 4 2 4
T(Γ(F ×F )) is not toroidal.
4 4
Finally,Figure1part(d)showsasubgraphofT(Γ(Z ×F ))thatisasubdivision
3 4
of K , and thus by Theorem 1.2 γ(T(Γ(Z ×F )))≥2. (cid:3)
5,4 3 4
The authors wish to thank the referee for the detailed and useful comments.
6 HAMIDREZAMAIMANI,CAMERONWICKHAM,ANDSIAMAKYASSEMI
Figure 1. Embeddings in the torus and a subgraph of T(Γ(Z ×F ))
3 4
00 13 00 1α2
u u u u
(cid:0) (cid:0)(cid:0)@A (cid:0) (cid:0)@A
(cid:0) (cid:0)(cid:0) A@ (cid:0) (cid:0) A@
10 u(cid:0) u(cid:0)(cid:0)H12 A @u10 10 (cid:0)u u(cid:0)1α A @u10
@ H A (cid:0) A
H
@ H A (cid:0) A
H
11 u @u02 HAu11 11 (cid:0)u uH0α Au11
(cid:0)@@ (cid:0)(cid:0) (cid:0)@H
H
(cid:0) @@ (cid:0)(cid:0) (cid:0) @ H
H
01 u(cid:0) 03@@(cid:0)(cid:0)u u01 01 u(cid:0) 0@α2u Hu01
@ (cid:0) (cid:0) @ (cid:0)
@ (cid:0) (cid:0) @ (cid:0)
@u(cid:0) u(cid:0) @u(cid:0) u
00 13 00 1α2
(a) Embedding of Z ×Z (b) Embedding of Z ×F
2 4 2 4
11 12 10 11 1α 1α2
u u uHPP Hu (cid:8)u (cid:16)(cid:8)(cid:16)u
(cid:0)(cid:1) (cid:0)(cid:1)@ @H(cid:0)P@(cid:8)H(cid:0)@(cid:16)(cid:8)(cid:0)
H(cid:8)P(cid:16)P(cid:16)H(cid:8)
(cid:0)(cid:1) (cid:0)(cid:1) @ @(cid:0)(cid:8)(cid:16)H@(cid:0)(cid:8)PH(cid:0)@
(cid:16)(cid:8)(cid:16) (cid:8)H PHP
22 u(cid:0) (cid:1) (cid:0) (cid:1) (cid:8)@u 22 (cid:16)(cid:8)(cid:0)u @(cid:8)(cid:0)u @H(cid:0)u @HPu
(cid:1) (cid:0) (cid:1) (cid:8)(cid:8)(cid:0) 20S 21 2α 2α2
(cid:1)(cid:0) (cid:1) (cid:8) (cid:0) S
(cid:8)
21 (cid:0)u(cid:1) (cid:8)(cid:1)u (cid:0)u u 21 S u u u
@ 01@ (cid:0) 10 SA01 (cid:1) 0(cid:17)α(cid:17) 0α2
@ @(cid:0)u SA (cid:1) (cid:17)
(cid:17)
0@2 (cid:0)u 00@u 20 SA(cid:17)(cid:1)u
(cid:0)
00
(cid:0)
(cid:0)u u
11 12
(c) Embedding of Z ×Z (d) A subgraph of T(Γ(Z ×F ))
3 3 3 4
that is a subdivision of K
5,4
RINGS WHOSE TOTAL GRAPHS HAVE GENUS AT MOST ONE 7
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HamidRezaMaimani,DepartmentofMathematics,ShahidRajaeeUniversity,Tehran,
Iran,andInstitute for Research inFundamentalSciences (IPM),TehranIran.
CameronWickham,MathematicsDepartment,MissouriStateUniversity,Springfield,
MO65897,USA.
Siamak Yassemi, Department of Mathematics, University of Tehran, Tehran, Iran,
andInstitute for Research in FundamentalSciences (IPM),Tehran Iran.
