Classification of the Toroidal Jacobson Graphs

Abstract

Let R be a finite commutative ring with nonzero identity and denote its Jacobson radical by J(R). The Jacobson graph of R is the graph in which the vertex set is $R\setminus J(R)$, and two distinct vertices x and y are adjacent if and only if $1-xy$ is not a unit in R. In this paper, up to isomorphism, we classify the rings R whose Jacobson graphs are toroidal.

1 Introduction and the Statement of the Main Result

Algebraic combinatorics is an area of mathematics which employs methods of abstract algebra in various combinatorial contexts and vice versa. Associating a graph to an algebraic structure is a research subject in this area and has attracted considerable attention. The research in this subject aims at exposing the relationship between algebra and graph theory and at advancing the application of one to the other. In fact, there are three major problems in this subject: (1) characterization of the resulting graphs, (2) characterization of the algebraic structures with isomorphic graphs, and (3) realization of the connections between the algebraic structures and the corresponding graphs. In 1988, Beck [6] introduced the idea of a zero-divisor graph of a commutative ring R with nonzero identity. He defined $\varGamma _0(R)$ to be the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if $xy=0$. He was mostly concerned with coloring of $\varGamma _0(R)$. Beck conjectured that $\chi (R)= \omega (R)$, where $\chi (R)$ and $\omega (R)$ denote, respectively, the chromatic number and the clique number of $\varGamma _0(R)$. Such graphs are called weakly perfect graphs. This investigation of coloring of a commutative ring was then continued by Anderson and Naseer in [1]. They gave a counterexample for the above conjecture of Beck. In [3], Anderson and Livingston proposed a different method of associating a zero-divisor graph to a commutative ring R, and according to them this gives a better illustration of the zero-divisor structure of the ring. They defined $\varGamma (R)$ to be the graph in which the vertex set consists of all the nonzero zero-divisors of R, and two distinct vertices x and y are adjacent if and only if $xy=0$. For a survey and recent results concerning zero-divisor graphs, we refer the reader to [2]. In literature, one can find a number of different types of graphs attached to rings or other algebraic structures. For a survey of recent results concerning graphs attached to rings, we refer the reader to [9].

The present paper deals with what is known as the Jacobson graph of a ring. Given an arbitrary finite commutative ring R with nonzero identity, its Jacobson graph $\mathfrak {J}_R$ is defined to be the graph in which the vertex set is $R\setminus J(R)$, and two distinct vertices x and y are adjacent if and only if $1-xy\notin U(R)$. Here, J(R) denotes the Jacobson radical of R and U(R) is the set of unit elements in R. Some of the properties of this graph have been studied in detail in [4]. The graphs in Fig. 1 are the Jacobson graphs of the rings indicated. Throughout the paper, in all the figures, we abbreviate the ordered pair (r, s) by rs.

Fig. 1

The Jacobson graphs of some specific rings

Let us recall some notions from graph theory. In the sequel, by a graph we mean a finite graph without loops or multiple edges. The genus of a graph G, denoted by $\gamma (G)$, is the smallest nonnegative integer g such that the graph G can be embedded on the surface obtained by attaching g handles to a sphere. The graphs of genus zero are called planar graphs and the graphs of genus one are called toroidal graphs. Rings whose Jacobson graphs are planar have been already classified (see [4, Theorem 4.3]), and in this paper, up to isomorphism, we classify the rings whose Jacobson graphs are toroidal. More precisely, we prove the following theorem which is the main result of this paper. Note that, in the sequel, $\mathbb {F}_4$ denotes the ring $\frac{\mathbb {Z}_2[x]}{(x^2+x+1)}$.

Theorem 1.1

Let R be a finite commutative ring with nonzero identity. Then the Jacobson graph $\mathfrak {J}_R$ is a toroidal graph if and only if R is isomorphic to one of $\mathbb {Z}_2\times \mathbb {Z}_5$, $\mathbb {Z}_2\times \mathbb {Z}_7$, $\mathbb {Z}_3\times \mathbb {Z}_4$, $\mathbb {Z}_3\times \mathbb {F}_4$, $\mathbb {Z}_3\times \frac{\mathbb {Z}_2[x]}{(x^2)}$, $\frac{\mathbb {Z}_4[x]}{(x^2+x+1)}$, or $\frac{\mathbb {F}_4[x]}{(x^2)}$.

We divide the proof of Theorem 1.1 into a series of lemmas which we state and prove them in Sects. 2, 3, and 4. Furthermore, for the convenience of the reader, we state without proof a few known results in the form of propositions which will be used in the proofs of the lemmas. We also recall some definitions and notations concerning graphs for later use. For unexplained terminology and notations in this paper we refer the reader to [7].

2 Some Jacobson Graphs with Genus One

Let us start the process of classification by looking at some toroidal Jacobson graphs. The following result of paper [4] characterizes the planar Jacobson graphs which will be used later frequently. Note that the ring $\mathbb {Z}_{2}\times \frac{\mathbb {Z}_2[x]}{(x^2)}$ is missed in the paper [4].

Proposition 2.1

([4], Theorem 4.3) Let R be a finite commutative ring with nonzero identity. Then the Jacobson graph $\mathfrak {J}_R$ is a planar graph if and only if either R is a field or it is isomorphic to one of $\mathbb {Z}_{4}$, $\mathbb {Z}_ {8}$, $\mathbb {Z}_{9}$, $\mathbb {Z}_{2}\times \mathbb {Z}_{2}$, $\mathbb {Z}_{2}\times \mathbb {Z}_{3}$, $\mathbb {Z}_{2}\times \mathbb {Z}_{4}$, $\mathbb {Z}_{2}\times \mathbb {F}_{4}$, $\mathbb {Z}_{2}\times \frac{\mathbb {Z}_2[x]}{(x^2)}$, $\mathbb {Z}_{3}\times \mathbb {Z}_{3}$, $\mathbb {Z}_{2}\times \mathbb {Z}_{2}\times \mathbb {Z}_{2}$, $\frac{\mathbb {Z}_2[x]}{(x^2)}$, $\frac{\mathbb {Z}_2[x]}{(x^3)}$, $\frac{\mathbb { Z}_3[x]}{(x^2)}$, $\frac{\mathbb {Z}_4[x]}{(2x,x^2)}$, $\frac{\mathbb {Z}_4[x]}{(2x,x^2-2)}$, or $\frac{\mathbb {Z}_2[x,y]}{(x,y)^2}$.

In the following lemmas of this section, we give some rings with toroidal Jacobson graphs.

Lemma 2.2

If $R=\mathbb {Z}_2\times \mathbb {Z}_5$, then $\gamma (\mathfrak {J}_R)=1$.

Proof

By Proposition 2.1, we have $\gamma (\mathfrak {J}_R)\ge 1$. But Fig. 2 gives an embedding of the Jacobson graph $\mathfrak {J}_R$ on a torus and so $\gamma (\mathfrak {J}_R)=1$. $\square$

Fig. 2

Embedding of the Jacobson graph of $\mathbb {Z}_2\times \mathbb {Z}_5$ on a torus

Lemma 2.3

If $R=\mathbb {Z}_2\times \mathbb {Z}_7$, then $\gamma (\mathfrak {J}_R)=1$.

Proof

By Proposition 2.1, we have $\gamma (\mathfrak {J}_R)\ge 1$. But Fig. 3 gives an embedding of the Jacobson graph $\mathfrak {J}_R$ on a torus and so $\gamma (\mathfrak {J}_R)=1$. $\square$

Fig. 3

Embedding of the Jacobson graph of $\mathbb {Z}_2\times \mathbb {Z}_7$ on a torus

Lemma 2.4

If $R=\mathbb {Z}_3\times \mathbb {Z}_4$, then $\gamma (\mathfrak {J}_R)=1$.

Proof

By Proposition 2.1, we have $\gamma (\mathfrak {J}_R)\ge 1$. But Fig. 4 gives an embedding of the Jacobson graph $\mathfrak {J}_R$ on a torus and so $\gamma (\mathfrak {J}_R)=1$. $\square$

Fig. 4

Embedding of the Jacobson graph of $\mathbb {Z}_3\times \mathbb {Z}_4$ on a torus

Lemma 2.5

If $R=\mathbb {Z}_3\times \mathbb {F}_4$, then $\gamma (\mathfrak {J}_R)=1$.

Proof

By Proposition 2.1, we have $\gamma (\mathfrak {J}_R)\ge 1$. But Fig. 5 gives an embedding of the Jacobson graph $\mathfrak {J}_R$ on a torus and so $\gamma (\mathfrak {J}_R)=1$. $\square$

Fig. 5

Embedding of the Jacobson graph of $\mathbb {Z}_3\times \mathbb {F}_4$ on a torus

Lemma 2.6

If $R=\mathbb {Z}_3\times \frac{\mathbb {Z}_2[x]}{(x^2)}$, then $\gamma (\mathfrak {J}_R)=1$.

Proof

By Proposition 2.1, we have $\gamma (\mathfrak {J}_R)\ge 1$. But Fig. 6 gives an embedding of the Jacobson graph $\mathfrak {J}_R$ on a torus and so $\gamma (\mathfrak {J}_R)=1$. $\square$

Fig. 6

Embedding of the Jacobson graph of $\mathbb {Z}_3\times \frac{\mathbb {Z}_2[x]}{(x^2)}$ on a torus

A graph G is said to be complete if there is an edge between every pair of distinct vertices in G. We denote the complete graph with n vertices by $K_n$. A bipartite graph is the one whose vertex set can be partitioned into two disjoint parts in such a way that the two end vertices of every edge lie in different parts. Among the bipartite graphs, the complete bipartite graph is the one in which two vertices are adjacent if and only if they lie in different parts. The complete bipartite graph, with parts of size m and n, is denoted by $K_{m,n}$.

We now state the following propositions which are needed for proof of Lemma 2.11.

Proposition 2.7

([4], Theorem 2.2) Let R be a finite commutative ring with nonzero identity. If R is local with maximal ideal $\mathfrak {m}$ and residue field $\mathbb {K}=R/\mathfrak {m}$, then the connected components of $\mathfrak {J}_R$ are isomorphic either to complete graph $K_{|\mathfrak {m}|}$ or to complete bipartite graph $K_{|\mathfrak {m}|,| \mathfrak {m}|}$. Moreover,

• (1) if $|\mathbb {K}|$ is odd, then $\mathfrak {J}_R$ has two complete components and $(|\mathbb {K}|-3)/2$ complete bipartite components, and

• (2) if $|\mathbb {K}|$ is even, then $\mathfrak {J}_R$ has one complete component and $(|\mathbb {K}|-2)/2$ complete bipartite components.

Proposition 2.8

([12], Theorem 6–38) If $n\ge 3$, then

$$\begin{aligned} \gamma (K_{n})=\left\lceil \frac{(n-3)(n-4)}{12}\right\rceil . \end{aligned}$$

Proposition 2.9

([12], Theorem 6–37) If $m,n\ge 2$, then

$$\begin{aligned} \gamma (K_{m,n})=\left\lceil \frac{(m-2)(n-2)}{4} \right\rceil . \end{aligned}$$

Given a graph G, we denote its vertex set by V(G) and its edge set by E(G). If $G_1$ and $G_2$ are any two graphs, then their disjoint union, denoted by $G_1 \sqcup G_2$, is defined to be the graph in which the vertex set is $V(G_1) \sqcup V(G_2)$ and the edge set is $E(G_1) \sqcup E(G_2)$. The following result, which follows from [5, Corollary 2], often enables us to reformulate some results which are otherwise true for connected graphs.

Proposition 2.10

If a graph G is isomorphic to the disjoint union $G_1 \sqcup G_2$ of two graphs $G_1$ and $G_2$, then $\gamma (G)=\gamma (G_1)+\gamma (G_2)$.

We are now ready to state and prove the final result of this section.

Lemma 2.11

If R is one of the rings $\frac{\mathbb {Z}_4[x]}{(x^2+x+1)}$ and $\frac{\mathbb {F}_4[x]}{(x^2)}$, then $\gamma (\mathfrak {J}_R)=1$.

Proof

By Proposition 2.7, the Jacobson graph $\mathfrak {J}_R$ is isomorphic to the disjoint union of a copy of $K_4$ and a copy of $K_{4,4}$. Hence, by Propositions 2.10, 2.8, and 2.9, we obtain that $\gamma (\mathfrak {J}_R)=1$. $\square$

3 Some Jacobson Graphs with Genus Greater than One

We continue the paper by looking at some rings whose Jacobson graphs have genus greater than one. Let us start with the following lemma.

Lemma 3.1

If R is one of the rings $\mathbb {Z}_{25}$ and $\frac{\mathbb {Z}_5[x]}{(x^2)}$, then $\gamma (\mathfrak {J}_R)=5$.

Proof

By Proposition 2.7, the Jacobson graph $\mathfrak {J}_R$ is isomorphic to the disjoint union of two copies of $K_5$ and a copy of $K_{5,5}$. Hence, by Propositions 2.10, 2.8, and 2.9, we obtain that $\gamma (\mathfrak {J}_R) =5$. $\square$

Lemma 3.2

If R is one of the rings $\mathbb {Z}_{49}$ and $\frac{\mathbb {Z}_7[x]}{(x^2)}$, then $\gamma (\mathfrak {J}_R)=16$.

Proof

By Proposition 2.7, the Jacobson graph $\mathfrak {J}_R$ is isomorphic to the disjoint union of two copies of $K_7$ and two copies of $K_{7,7}$. Hence, by Propositions 2.10, 2.8, and 2.9, we obtain that $\gamma (\mathfrak {J}_R)=16$. $\square$

Lemma 3.3

If R is one of the rings $\mathbb {Z}_3\times \mathbb {Z}_5$ and $\mathbb { Z}_3\times \mathbb {Z}_7$, then $\gamma (\mathfrak {J}_R)\ge 2$.

Proof

First, suppose that $R=\mathbb {Z}_3\times \mathbb {Z}_5$. Consider the following two subsets $V_1$ and $V_2$ of the vertex set of $\mathfrak {J}_R$:

$$\begin{aligned} V_1= & {} \{(1,0),(1,1),(1,2),(1,3),(1,4)\},\\ V_2= & {} \{(2,0),(2,1),(2,2),(2,3),(2,4)\}. \end{aligned}$$

It is easy to see that the two subgraphs $\langle V_1\rangle$ and $\langle V_2\rangle$ are disjoint, and $\langle V_1\rangle \cong \langle V_2\rangle \cong K_5$. Therefore, it follows from Propositions 2.10 and 2.8 that $\gamma (\mathfrak {J}_R)\ge 2$.

Second, suppose that $R=\mathbb {Z}_3\times \mathbb {Z}_7$. Consider the following two subsets $V_3$ and $V_4$ of the vertex set of $\mathfrak {J}_R$:

$$\begin{aligned} V_3= & {} \{(1,0),(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)\},\\ V_4= & {} \{(2,0),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)\}. \end{aligned}$$

It is easy to see that the two subgraphs $\langle V_3\rangle$ and $\langle V_4\rangle$ are disjoint, and $\langle V_3\rangle \cong \langle V_4\rangle \cong K_7$. Therefore, it follows from Propositions 2.10 and 2.8 that $\gamma (\mathfrak {J}_R)\ge 2$. $\square$

Lemma 3.4

If R is one of the rings $\mathbb {F}_4\times \mathbb {Z}_5$ and $\mathbb {Z} _5\times \mathbb {Z}_5$, then $\gamma (\mathfrak {J}_R)\ge 3$.

Proof

Consider the following two subsets $V_1$ and $V_2$ of the vertex set of $\mathfrak {J}_R$. If $R=\mathbb {F}_4\times \mathbb {Z}_5$ is the case, we may consider

$$\begin{aligned} V_1= & {} \{(x,0),(x,1),(x,2),(x,3),(x,4)\},\\ V_2= & {} \{(x+1,0),(x+1,1),(x+1,2),(x+1,3),(x+1,4)\}, \end{aligned}$$

and if $R=\mathbb {Z}_5\times \mathbb {Z}_5$ is the case, we may consider

$$\begin{aligned} V_1= & {} \{(2,0),(2,1),(2,2),(2,3),(2,4)\},\\ V_2= & {} \{(3,0),(3,1),(3,2),(3,3),(3,4)\}. \end{aligned}$$

In each cases, it is easy to see that $\langle V_1\cup V_2\rangle$ has a subgraph isomorphic to $K_{5,5}$. Therefore, it follows from Propositions 2.10 and 2.9 that $\gamma (\mathfrak {J}_R)\ge 3$. $\square$

Lemma 3.5

If R is one of the rings $\mathbb {F}_4\times \mathbb {Z}_7$, $\mathbb {Z} _5\times \mathbb {Z}_7$, and $\mathbb {Z}_7\times \mathbb {Z}_7$, then $\gamma (\mathfrak {J}_R)\ge 7$.

Proof

Consider the following two subsets $V_1$ and $V_2$ of the vertex set of $\mathfrak {J}_R$. If $R=\mathbb {F}_4\times \mathbb {Z}_7$ is the case, we may consider

$$\begin{aligned} V_1= & {} \{(x,0),(x,1),(x,2),(x,3),(x,4),(x,5),(x,6)\},\\ V_2= & {} \{(x\!+\!1,0),(x\!+\!1,1),(x\!+\!1,2),(x+1,3),(x+1,4),(x+1,5),(x+1,6)\}, \end{aligned}$$

if $R=\mathbb {Z}_5\times \mathbb {Z}_7$ is the case, we may consider

$$\begin{aligned} V_1= & {} \{(2,0),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)\},\\ V_2= & {} \{(3,0),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)\}, \end{aligned}$$

and finally, if $R=\mathbb {Z}_7\times \mathbb {Z}_7$ is the case, we may consider

$$\begin{aligned} V_1= & {} \{(2,0),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)\},\\ V_2= & {} \{(4,0),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)\}. \end{aligned}$$

In each cases, it is easy to see that $\langle V_1\cup V_2\rangle$ has a subgraph isomorphic to $K_{7,7}$. Therefore, it follows from Propositions 2.10 and 2.9 that $\gamma (\mathfrak {J}_R)\ge 7$. $\square$

Let x and y be any two vertices in a graph G. A path between x and y is a sequence $\{x,x_1\},\{x_1,x_2\},\ldots ,\{x_n, y\}$ of distinct edges, which is also written as $\{x,x_1,x_2,\ldots ,x_n,y\}$, where the vertices $x,x_1, x_2,\ldots ,x_n,y$ are all distinct (except, possibly, x and y). A path between x and y is called a cycle if $x=y$. The number of edges in a path or a cycle is called its length.

A graph G is said to be connected if there is a path between every pair of distinct vertices in G. A chord of a cycle in a graph is an edge of the graph which does not lie in the edge set of the cycle but whose endpoints lie in the vertex set of the cycle. A chordless cycle of a graph is a cycle without any chord.

A cycle of a graph, embedded on a surface, is called contractible with respect to the embedding if it can be contracted continuously on the surface to a point. A cycle of a toroidal graph is said to be flat if it is contractible in every torus embedding of the graph. Given a cycle C of a graph G, we write $G-C$ to denote the graph obtained from G by deleting the vertices of C and the edges of the graph incident to the vertices of C.

A subdivision of an edge $\{x,y\}$ in a graph is a path $\{x,x_1,x_2, \ldots ,x_n,y\}$ obtained by inserting some new vertices $x_1$, $x_2$, ..., $x_n$ into the edge $\{x,y\}$. A subdivision of a graph G is the result of some subdivisions of the edges of G. Furthermore, every graph can be considered as a subdivision of itself. A remarkably simple characterization of planar graphs was given by Kuratowski in 1930. Kuratowski’s theorem [7, p. 153] says that a graph is planar if and only if it contains no subdivision of $K_{3,3}$ or $K_5$. As a consequence of Kuratowski’s theorem, one has the following result.

Proposition 3.6

([11], Theorem 2.1) If a cycle C of a toroidal graph G is such that $G-C$ is nonplanar, then C is flat in G. Furthermore, if flat C is chordless and $G-C$ is connected, then C is a flat face in any torus embedding of G.

Lemma 3.7

If $R=\mathbb {F}_4\times \mathbb {F}_4$, then $\gamma (\mathfrak {J}_R)\ge 2$.

Proof

By Proposition 2.1, we have $\gamma (\mathfrak {J}_R)\ge 1$. Let us now assume that the Jacobson graph $\mathfrak {J}_{R}$ is toroidal, that is, $\gamma (\mathfrak {J}_R)=1$. We use Proposition 3.6 to show that the following 28 cycles in $\mathfrak {J}_R$ are all faces of the graph $\mathfrak {J}_R$, and arrive at a contradiction. In this direction, consider the following cycles in $\mathfrak {J}_R$:

$$\begin{aligned} C_1= & {} \{(0,x),(0,x+1),(1,x),(1,x+1),(0,x)\}, \\ C_2= & {} \{(0,x),(0,x+1),(1,x),(x,x+1),(0,x)\}, \\ C_3= & {} \{(0,x),(0,x+1),(1,x),(x+1,x+1),(0,x)\}, \\ C_4= & {} \{(0,x),(x,x+1),(x+1,x+1),(0,x)\}, \\ C_5= & {} \{(0,x+1),(0,x),(1,x+1),(x,x),(0,x+1)\}, \\ C_6= & {} \{(0,x+1),(0,x),(1,x+1),(x+1,x),(0,x+1)\}, \\ C_7= & {} \{(0,x+1),(x,x),(x+1,x),(0,x+1)\}, \\ C_8= & {} \{(1,0),(1,1),(1,x),(1,0)\}, \\ C_9= & {} \{(1,0),(1,1),(1,x+1),(1,0)\}, \\ C_{10}= & {} \{(1,0),(1,x),(1,x+1),(1,0)\}, \\ C_{11}= & {} \{(1,x),(1,x+1),(x,x),(0,x+1),(1,x)\}, \\ C_{12}= & {} \{(1,x),(x,x+1),(x+1,x+1),(1,x)\}, \\ C_{13}= & {} \{(1,x+1),(1,x),(x+1,x+1),(0,x),(1,x+1)\}, \\ C_{14}= & {} \{(1,x+1),(x,x),(x+1,x),(1,x+1)\}. \end{aligned}$$

Let $1\le n\le 14$ be given and consider the following subsets $V_1$ and $V_2$ of the vertex set of $\mathfrak {J}_R$:

$$\begin{aligned} V_1= & {} \{(x,0),(x,1),(x,x),(x,x+1)\}, \\ V_2= & {} \{(x+1,0),(x+1,1),(x+1,x),(x+1,x+1)\}. \end{aligned}$$

It is easy to see that $G-C_n$ contains a subgraph of $\langle V_1\cup V_2\rangle$ which is isomorphic to $K_{3,3}$ and so it is nonplanar. Moreover, $C_n$ is chordless and $G-C_n$ is connected. Therefore, by Proposition 3.6, $C_n$ is a face.

Now, consider the following cycles in $\mathfrak {J}_R$:

$$\begin{aligned} C_{15}= & {} \{(0,1),(1,1),(x,1),(0,1)\}, \\ C_{16}= & {} \{(0,1),(1,1),(x+1,1),(0,1)\}, \\ C_{17}= & {} \{(0,1),(x,1),(x+1,1),(0,1)\}, \\ C_{18}= & {} \{(x,0),(x+1,0),(x,1),(x+1,1),(x,0)\}, \\ C_{19}= & {} \{(x,0),(x+1,0),(x,1),(x+1,x),(x,0)\}, \\ C_{20}= & {} \{(x,0),(x+1,0),(x,1),(x+1,x+1),(x,0)\}, \\ C_{21}= & {} \{(x,0),(x+1,x),(x+1,x+1),(x,0)\}, \\ C_{22}= & {} \{(x,1),(x+1,1),(x,x),(x+1,0),(x,1)\}, \\ C_{23}= & {} \{(x,1),(x+1,x),(x+1,x+1),(x,1)\}, \\ C_{24}= & {} \{(x+1,0),(x,0),(x+1,1),(x,x),(x+1,0)\}, \\ C_{25}= & {} \{(x+1,0),(x,0),(x+1,1),(x,x+1),(x+1,0)\}, \\ C_{26}= & {} \{(x+1,0),(x,x),(x,x+1),(x+1,0)\}, \\ C_{27}= & {} \{(x+1,1),(x,1),(x+1,x+1),(x,0),(x+1,1)\}, \\ C_{28}= & {} \{(x+1,1),(x,x),(x,x+1),(x+1,1)\}. \end{aligned}$$

Let $15\le n\le 28$ be given and consider the following subsets $V_3$ and $V_4$ of the vertex set of $\mathfrak {J}_R$:

$$\begin{aligned} V_3= & {} \{(0,x),(1,x),(x,x),(x+1,x)\}, \\ V_4= & {} \{(0,x+1),(1,x+1),(x,x+1),(x+1,x+1)\}. \end{aligned}$$

Again, $G-C_n$ contains a subgraph of $\langle V_3\cup V_4\rangle$ which is isomorphic to $K_{3,3}$ and so it is nonplanar. Moreover, $C_n$ is chordless and $G-C_n$ is connected. Therefore, by Proposition 3.6, $C_n$ is a face.

All in all, in this stage, the Jacobson graph $\mathfrak {J}_R$ has at least 28 faces. But, it is easy to see that $\mathfrak {J}_R$ has 15 vertices and 40 edges, so by Euler’s formula, $\mathfrak {J}_{R}$ has only $40-15=25$ faces. This contradiction shows that $\gamma (\mathfrak {J}_R)\ge 2$. $\square$

Lemma 3.8

If $R=\mathbb {Z}_2\times \mathbb {Z}_2\times \mathbb {Z}_3$, then $\gamma (\mathfrak {J}_R)\ge 2$.

Proof

Since $\mathbb {Z}_2\times \mathbb {Z}_2\times \mathbb {Z}_3\cong \mathbb {Z}_2 \times \mathbb {Z}_6$, it is enough to prove the lemma for $R=\mathbb {Z}_2 \times \mathbb {Z}_6$. By Proposition 2.1, we have $\gamma (\mathfrak {J}_R) \ge 1$. Let us now assume that the Jacobson graph $\mathfrak {J}_{R}$ is toroidal, that is, $\gamma (\mathfrak {J}_R)=1$. We use Proposition 3.6 to show that the following 26 cycles in $\mathfrak {J}_R$ are all faces of the graph $\mathfrak {J}_R$, and arrive at a contradiction. In this direction, consider the following cycles in $\mathfrak {J}_R$:

$$\begin{aligned} C_1= & {} \{(0,1),(0,3),(0,5),(0,1)\}, \\ C_2= & {} \{(0,1),(0,3),(1,1),(0,1)\}, \\ C_3= & {} \{(0,1),(0,3),(1,3),(0,1)\}, \\ C_4= & {} \{(0,1),(0,3),(1,5),(0,1)\}, \\ C_5= & {} \{(0,1),(0,4),(1,1),(0,1)\}, \\ C_6= & {} \{(0,1),(0,4),(1,4),(0,1)\}, \\ C_7= & {} \{(0,1),(0,5),(1,1),(0,1)\}, \\ C_8= & {} \{(0,1),(0,5),(1,3),(0,1)\}, \\ C_9= & {} \{(0,1),(0,5),(1,5),(0,1)\}, \\ C_{10}= & {} \{(0,2),(0,5),(1,2),(0,2)\}, \\ C_{11}= & {} \{(0,2),(0,5),(1,5),(0,2)\}, \\ C_{12}= & {} \{(0,3),(0,5),(1,1),(0,3)\}, \\ C_{13}= & {} \{(0,3),(0,5),(1,3),(0,3)\}, \\ C_{14}= & {} \{(0,3),(0,5),(1,5),(0,3)\}. \end{aligned}$$

Let $1\le n\le 14$ be given and consider the following subset $V_1$ of the vertex set of $\mathfrak {J}_R$:

$$\begin{aligned} V_1=\{(1,0),(1,1),(1,2),(1,3),(1,4),(1,5)\}. \end{aligned}$$

It is easy to see that $G-C_n$ contains a subgraph of $\langle V_1\rangle$ which is isomorphic to $K_5$ and so it is nonplanar. Moreover, $C_n$ is chordless and $G-C_n$ is connected. Therefore, by Proposition 3.6, $C_n$ is a face.

Now, consider the following cycles in $\mathfrak {J}_R$:

$$\begin{aligned} C_{15}= & {} \{(0,2),(1,2),(1,5),(0,2)\}, \\ C_{16}= & {} \{(0,4),(1,1),(1,4),(0,4)\}, \\ C_{17}= & {} \{(1,0),(1,1),(1,2),(1,0)\}, \\ C_{18}= & {} \{(1,0),(1,1),(1,4),(1,0)\}, \\ C_{19}= & {} \{(1,0),(1,2),(1,3),(1,0)\}, \\ C_{20}= & {} \{(1,0),(1,2),(1,4),(1,0)\}, \\ C_{21}= & {} \{(1,0),(1,2),(1,5),(1,0)\}, \\ C_{22}= & {} \{(1,0),(1,3),(1,4),(1,0)\}, \\ C_{23}= & {} \{(1,0),(1,4),(1,5),(1,0)\}, \\ C_{24}= & {} \{(1,1),(1,2),(1,4),(1,1)\}, \\ C_{25}= & {} \{(1,2),(1,3),(1,4),(1,2)\}, \\ C_{26}= & {} \{(1,2),(1,4),(1,5),(1,2)\}. \end{aligned}$$

Let $15\le n\le 26$ be given and consider the following subset $V_2$ of the vertex set of $\mathfrak {J}_R$:

$$\begin{aligned} V_2=\{(0,1),(0,3),(0,5),(1,1),(1,3),(1,5)\}. \end{aligned}$$

Again, $G-C_n$ contains a subgraph of $\langle V_2\rangle$ which is isomorphic to $K_5$ and so it is nonplanar. Moreover, $C_n$ is chordless and $G-C_n$ is connected. Therefore, by Proposition 3.6, $C_n$ is a face.

All in all, in this stage, the Jacobson graph $\mathfrak {J}_R$ has at least 26 faces. But, it is easy to see that $\mathfrak {J}_{R}$ has 11 vertices and 35 edges, so by Euler’s formula, $\mathfrak {J}_{R}$ has only $35-11=24$ faces. This contradiction shows that $\gamma (\mathfrak {J}_R)\ge 2$. $\square$

4 Completing the Proof of Theorem 1.1

In this section, we complete the proof of Theorem 1.1. For this purpose, we need some more results.

Lemma 4.1

Let R be a finite commutative ring with nonzero identity having a maximal ideal of size $\ge 8$. Then $\gamma (\mathfrak {J}_R)\ge 2$.

Proof

By the assumption, R has at least a maximal ideal $\mathfrak {m}=\{m_1,\ldots , m_{\ell }\}$ with $\ell \ge 8$. Let $i\ne j$ with $1\le i,j\le \ell$ be given. Note that $1+m_i$ and $1+m_j$ are distinct elements of $R\setminus J(R)$, and $1-(1+m_i)(1+m_j)\notin U(R)$. Therefore, $1+m_i$ and $1+m_j$ are adjacent vertices in $\mathfrak {J}_R$. This implies that the vertices $1+m_1,\ldots ,1+m_ {\ell }$ are mutually adjacent in the Jacobson graph $\mathfrak {J}_R$ and so it has a subgraph isomorphic to $K_{\ell }$, where $\ell \ge 8$. Now, Proposition 2.8 implies that $\gamma (\mathfrak {J}_R)\ge 2$. $\square$

We also need to characterize all finite commutative rings with nonzero identity which are either a local ring, a product of two local rings or a product of three local rings, and all of their maximal ideals have size $\le$7. In the following three lemmas we give this characterization.

Lemma 4.2

Let R be a finite commutative ring with nonzero identity such that its all maximal ideals have size $\le$7. If R is a local ring, then it is either a field or is isomorphic to one of the rings given by Table 1.

Table 1 Finite commutative local rings R with maximal ideals $\mathfrak {m}$ of size $\le$7

Proof

The lemma may be obtained by using [8] together with some known results on the structures of small local rings. $\square$

Lemma 4.3

Let R be a finite commutative ring with nonzero identity such that its all maximal ideals have size $\le$7. If R is a product of two local rings, then it is isomorphic to one of the rings given by Table 2.

Table 2 Finite commutative rings R with maximal ideals $\mathfrak {m}$ of size $\le$7 which are a product of two local rings

Proof

By the assumption, R is a product of two local rings, say $R_1$ and $R_2$. Let $\mathfrak {m}_1$ and $\mathfrak {m}_2$ be the maximal ideals of $R_1$ and $R_2$, respectively.

First, we may assume that none of $R_1$ and $R_2$ is a field. In this case, for $i=1,2$, we have $|R_i|\ge 4$ and $|\mathfrak {m}_i|\ge 2$. Note that $\mathfrak {m}_1\times R_2$ is a maximal ideal of R with $|\mathfrak {m}_1\times R_2|\ge 8$, a contradiction. Hence, this is not the case and so either both of $R_1$ and $R_2$ are fields or only one of them is a field.

Case 1: Both of $R_1$ and $R_2$ are fields. In this case, $\{0\} \times R_2$ and $R_1\times \{0\}$ are the maximal ideals of R. Now, by the assumption, for $i=1,2$, we obtain that $|R_i|\le 7$. Therefore, R is isomorphic to one of the following rings:

Case 2: Only, say $R_1$, is a field. In this case, $R_2$ is not a field and so we have $|R_2|\ge 4$ and $|\mathfrak {m}_2|\ge 2$. Note that $\{0\}\times R_2$ and $R_1\times \mathfrak {m}_2$ are the maximal ideals of R. Now, by the assumption, $|R_1|\le 3$ and $4\le |R_2|\le 7$. Therefore, R is isomorphic to one of $\mathbb {Z}_2\times \mathbb {Z}_4$, $\mathbb {Z} _2\times \frac{\mathbb {Z}_2[x]}{(x^2)}$, $\mathbb {Z}_3\times \mathbb {Z}_4$, or $\mathbb {Z}_3\times \frac{\mathbb {Z}_2[x]}{(x^2)}$. $\square$

Lemma 4.4

Let R be a finite commutative ring with nonzero identity such that its all maximal ideals have size $\le$7. If R is a product of three local rings, then it is isomorphic to one of $\mathbb {Z}_2\times \mathbb {Z}_2 \times \mathbb {Z}_2$ and $\mathbb {Z}_2\times \mathbb {Z}_2\times \mathbb {Z}_3$.

Proof

By the assumption, R is a product of three local rings, say $R_1$, $R_2$, and $R_3$. If one of the $R_i$s has size $\ge 4$, then R has a maximal ideal of size $\ge 8$, a contradiction. If two of the $R_i$s have size equal to 3, then R has a maximal ideal of size $\ge 9$, again a contradiction. Therefore, R is isomorphic to one of $\mathbb {Z}_2\times \mathbb {Z}_2 \times \mathbb {Z}_2$ and $\mathbb {Z}_2\times \mathbb {Z}_2\times \mathbb {Z} _3$.

$\square$

Let us now summarize what we have achieved so far. By using Lemmas 4.2, 4.3, and 4.4 together Proposition 2.1 and all lemmas of Sects. 2 and 3 one has Table 3.

Table 3 Genus of all of the Jacobson graphs arising from finite commutative rings R with nonzero identity which are either a local ring, a product of two local rings or a product of three local rings, and all of their maximal ideals have size $\le$7

We are now in the position to give a proof of Theorem 1.1.

Proof of Theorem 1.1

($\Rightarrow$): Suppose that R is a finite commutative ring with nonzero identity such that its Jacobson graph is toroidal. By [10, p. 95], we may write $R\cong R_1\times \cdots \times R_{\ell }$, where every $R_i$ is a local ring with maximal ideal $\mathfrak {m}_i$. Since $\gamma (\mathfrak {J}_R)=1$, by Lemma 4.1, the size of all of the maximal ideals of R is at most 7. This forces that $\ell \le 3$, that is, either R is a local ring, R is a product of two local rings, or is a product of three local rings. Now, Table 3 implies that R is isomorphic to one of $\mathbb {Z}_2\times \mathbb {Z}_5$, $\mathbb {Z}_2\times \mathbb {Z}_7$, $\mathbb {Z}_3\times \mathbb {Z}_4$, $\mathbb {Z}_3\times \mathbb {F}_4$, $\mathbb {Z}_3\times \frac{\mathbb {Z}_2[x]}{(x^2)}$, $\frac{\mathbb {Z}_4[x]}{(x^2+x+1)}$, or $\frac{\mathbb {F}_ 4[x]}{(x^2)}$.

($\Leftarrow$): If R is isomorphic to one of $\mathbb {Z}_2\times \mathbb {Z}_5$, $\mathbb {Z}_2\times \mathbb {Z}_7$, $\mathbb {Z}_3\times \mathbb {Z}_4$, $\mathbb {Z}_3\times \mathbb {F}_4$, $\mathbb {Z}_3\times \frac{ \mathbb {Z}_2[x]}{(x^2)}$, $\frac{\mathbb {Z}_4[x]}{(x^2+x+1)}$, or $\frac{\mathbb {F}_ 4[x]}{(x^2)}$, then again by Table 3 we obtain that $\gamma (\mathfrak {J}_R)=1$. $\square$