Abstract
Recently, Katre et al. introduced the concept of the coprime index of a graph. They asked to characterize the graphs for which the coprime index is the same as the clique number. In this paper, we partially solve this problem. In fact, we prove that the clique number and the coprime index of a zero-divisor graph of an ordered set and the zero-divisor graph of a ring coincide. Also, it is proved that the annihilating ideal graphs, the co-annihilating ideal graphs and the comaximal ideal graphs of commutative rings can be realized as the zero-divisor graphs of specially constructed posets. Hence the coprime index and the clique number coincide for these graphs as well.
1 Introduction
Graph labeling is a very active area of research. Many researchers studied several types of graph labelings. For an excellent survey on graph labelings, we refer to Gallian [Citation12]. Recently, Katre et al. [Citation16] studied another labeling known as coprime labeling.
Definition 1.1.
(Katre et al. [Citation16]) Let be a graph of order n. An injection is called a coprime labeling of G if for any two vertices , u and v are adjacent if and only if f(u) and f(v) are coprime. A graph that admits a coprime labeling is called a coprime graph. A prime number p is said to be used by the coprime labeling f if p divides f(v) for some . Let be the number of primes used by the coprime labeling f. Then is called the coprime index of G.
It is known that every graph G is a coprime graph (cf. [Citation16, Theorem 2.3]). Further, the coprime index of a graph G is nothing but the edge clique covering number of the complement of G (cf. [Citation16, Theorem 2.10]). Note that finding the edge clique covering number of a graph G is NP-complete; see [Citation13]. By an edge clique cover of a graph G, we mean a collection of cliques such that . The minimum cardinality of an edge clique cover of G is called the edge clique covering number of G and is denoted by . The best bound for is due to Erdös, Goodman, and Posa [Citation11] which states that if G is any graph of order n, then , and this bound is attained.
The edge clique covering number of a graph G is in fact the intersection number of G, that is, ; see Fred Roberts [Citation21, Theorem 5]. The intersection number of G, denoted by i(G), is the minimum cardinality of a set X such that G is the intersection graph of a family of subsets of X.
Looking at the relations between the coprime index and the edge clique covering number, it is clear that finding the coprime index of a graph is a tough problem.
The clique number of a graph G is the size of the largest complete subgraph of G. The chromatic number of a graph G is the minimum number of colors required to color the vertices of G such that no two adjacent vertices receive the same color.
The following are some observations.
Observation 1.2.
[16, Observations 2.6, 2.7, 2.8]
.
If H is an induced subgraph of a graph G, then .
.
In view of Observation 1.2 (3), Katre et al. [Citation16] raised the following problem.
Problem 1.3.
[16, Problem 2.9] Characterize graphs G for which .
In this paper, we partially solve this problem. In fact, we prove that the clique number and the coprime index of a zero-divisor graph of an ordered set and the zero-divisor graph of a ring coincide. Also, it is proved that the annihilating ideal graphs, the co-annihilating ideal graphs and the comaximal ideal graphs of commutative rings can be realized as the zero-divisor graphs of specially constructed posets. Hence the coprime index and the clique number coincide for these graphs as well.
In the first result, we prove that the coprime index of a graph and its reduced graph remain the same. The reduced graph usually has less number of vertices. So this technique is useful and effective.
Definition 1.4.
[Citation7, Definition 2.1] Let G be a simple graph. Consider an equivalence relation ∼ on G: if and only if , where . The reduced graph of G, denoted by Gred, is the simple graph whose vertex set is , and two distinct equivalence classes and are adjacent in Gred if and only if u and v are adjacent in G.
A graph G and its reduced graph Gred are shown in . It is easily observed that Gred is isomorphic to an induced subgraph of G. Henceforth, we abuse the notation and will say that Gred is an induced subgraph of G. Clearly, if the graphs G and H are isomorphic, then .
Theorem 1.5.
Let G be a finite simple graph of order n and Gred be its reduced graph. Then .
Proof.
Since Gred is an induced subgraph of G, . Hence, it is enough to prove that . For this, let , where be a coprime labeling of Gred. Now, we define the coprime labeling of G using f. Let be the k distinct vertices of Gred. Clearly, . Let . Therefore, and .
Define a map by . Clearly, is an injective map, as f is injective. It is easy to observe that yij and yrs are adjacent in G if and only if and are adjacent in Gred if and only if and are coprime (as f is a coprime labeling of Gred) if and only if and are coprime if and only if and are coprime. This proves is a coprime labeling of G. Thus, . Hence, . □
Proposition 1.6.
[Citation7, Proposision 2.4] G is a complete r-partite graph if and only if Gred is the complete graph Kr.
In [Citation16], Katre et al. observed that, . In view of Theorem 1.5 and Proposition 1.6, we have the following corollary.
Corollary 1.7.
Let G be a complete r-partite graph. Then .
In [Citation16], Katre et al. observed that the clique number is less than or equal to the coprime index of G. We sharpen this lower bound of the coprime index in the following result.
Theorem 1.8.
Let G be a finite simple graph. Then .
Proof.
It is well known that . We want to prove that . Let and let be the m distinct primes used by the coprime labeling f of G. To prove , it is enough to prove that . For this, define the set , and the set for . Clearly, . We assign m different colors to these Vi’s. Now, we prove that, this is a proper coloring of G. For this, if x and y are adjacent in G, then f(x) and f(y) are coprime (as f is a coprime labeling of G). This implies x and y cannot be contained in the same Vi. So x and y have different colors. Hence, this is a proper coloring with m colors. Thus, . This proves . □
Definition 1.9.
(Anderson and Livingston [Citation5]) Let R be a commutative ring with identity. The zero-divisor graph of R is a simple graph with the vertex set , the set of non-zero zero-divisors of R, and two distinct vertices x, y are adjacent if and only if xy = 0.
Notation 1.10.
(Pirzada et al. [Citation20]) Let p be a prime number. Consider the partition of the vertex set into , where . Then .
Rewrite the elements of Vi as , where . Then the elements and are adjacent in if and only if . Also, the elements and are adjacent in the complement graph if and only if . The zero-divisor graphs and their complements for n = 8 and n = 7 are shown in . In the figures, a loop at Vi denotes that any two vertices of Vi are adjacent, that is, the vertices of Vi induce a complete graph on vertices and Vi is adjacent to Vj denotes that each vertex of Vi is adjacent to every vertex of Vj, that is, is a graph join. The join G + H of two graphs G and H is a graph formed from disjoint copies of G and H by connecting each vertex of G to each vertex of H.
We illustrate the coprime labelings of and in . The primes used in the coprime labeling of are p21, p22 and p31. Hence, . Also, the primes used in the coprime labeling of are and p41. Therefore, . The following result is essentially proved in [Citation8, Proposition 2.3] and [Citation20, Theorem 2].
Theorem 1.11.
([Citation8, Proposition 2.3], [Citation20, Theorem 2]) Let be the zero-divisor graph of a ring , where p is a prime. Then
With this preparation, we prove our first main result.
Theorem 1.12.
Let be the zero-divisor graph of a ring , where p is a prime. Then
Proof.
Consider the partition of as defined in Notation 1.10.
Let , where . We have the following two main cases on the values of n.
Case A: Let n be an even number.
Consider a map as follows: where pij are distinct primes. Now, we prove that f is a coprime labeling of . For this, let be distinct vertices of . We study the following three sub-cases.
Sub-case A-I: If for some , then , for some (see Notation 1.10). Clearly, . This implies that u and v are not adjacent in . Clearly, the primes divide both and , for . Therefore, f(u) and f(v) are not coprime.
Sub-case A-II: If for some , then , for some . Clearly, . This gives u and v are adjacent in . By the definition of f, and . As u and v are distinct vertices, . This implies f(u) and f(v) are coprime.
Sub-case A-III: If for some , then for some . In this case, and are adjacent if and only if . If , then . By the definition of f, the primes does not divide , for all . In particular, the prime does not divide . As , we have and are coprime. Hence, f(u) and f(v) are coprime. If , then u and v are not adjacent and (). By the definition of f, divides both and . Therefore, f(u) and f(v) are not coprime.
Thus, f is a coprime labeling of .
The number of distinct primes used in the coprime labeling . This implies . Therefore, by Theorem 1.8, .
Case B: Let n be an odd number.
Consider a map as follows: where pij and are distinct primes. Now, we prove that f is a coprime labeling of . For this, let be distinct vertices of . We study the following three sub-cases.
Sub-case B-I: If for some , then for some . Clearly, . Hence u and v are not adjacent in . Clearly, the prime divides both and . Therefore, f(u) and f(v) are not coprime.
Sub-case B-II: If for some , then for some . Clearly, . Hence u and v are adjacent in . By the definition of f, and . As u and v are distinct vertices, . This implies f(u) and f(v) are coprime.
Sub-case B-III: If for some , then , for some . In this case, and are adjacent if and only if . If , then . By the definition of f, the primes does not divide , for all . In particular, the prime does not divide . As , we have and are coprime. Hence, f(u) and f(v) are coprime. If , then u and v are not adjacent and (). By the definition of f, divides both and . Therefore, f(u) and f(v) are not coprime.
Thus, f is a coprime labeling of .
The number of distinct primes used in the coprime labeling .
This implies that . Therefore, by Theorem 1.8, . □
Definition 1.13.
[Citation1, Definition 2.1] An independent set in a graph G is a subset I of the vertex set V(G) of G such that no two vertices of I are adjacent. The independence number of G, denoted by , is defined as the cardinality of a maximum independent set of G.
Theorem 1.14.
[Citation1, Theorem 2] Let be the zero-divisor graph of a ring , where p is a prime. Then
The details of the proof of Theorem 1.15 can be obtained on similar lines as that of Theorem 1.11. Hence we skipped the detailed proof. The coprime labeling is given.
Theorem 1.15.
Let be the complement of the zero-divisor graph of a ring , where p is a prime. Then
Proof.
Case A: Let n be an even number.
Consider a map as follows: where pij and are distinct primes. Then, f is a coprime labeling of .
The number of distinct primes used in the coprime labeling
Case B: Let n be an odd number.
Consider a map as follows: where pij are distinct primes. Then, f is a coprime labeling of .
The number of distinct primes used in the coprime labeling □
Remark 1.16.
(1) Note that may not be true, if . In fact, it is not true for . Consider the ring . The zero-divisor graph and its reduced graph are shown in . Clearly, . Define the function such that , where are distinct primes. It is easy to verify that f is a coprime labeling of and . Hence, by Theorem 1.5, . Thus, .
(2) Also, may not be true, if . Consider the same example. Clearly, . It is easy to check that . Therefore, . Thus, .
(3) Using similar techniques used in Theorems 1.12 and 1.15, it can be proved that the coprime index and the clique number coincide for the zero-divisor graph and the complement of the zero-divisor graph of a finite commutative special principal ideal ring.
We need the following definitions given in [Citation10].
Let P be a poset. Given any , the upper cone of A is given by for every and the lower cone of A is given by for every . If , then the sets and will be denoted by a and , respectively. By , we mean . Dually, we have the notion of .
Suppose that P is a poset with 0. If , then the annihilator of A is given by , and if , then we write . An element is an atom if a > 0 and , and P is called atomic if for every , there exists an atom such that .
The direct product of posets is the poset with defined such that in P if and only if (in Pi) for every , where . For any , note that for every and . Similarly, for every and .
Let , where Pi’s are finite bounded posets such that . Since Pi’s are finite bounded poset, Pi’s are atomic. Note that if and only if Pi has the unique atom. Further, assume that is the unique atom Pi for every i. It is not difficult to prove that are the only n atoms of .
A poset P is called bounded if P has both the least element 0 and the greatest element 1.
Let P be a poset with 0. Define a zero-divisor of P to be any element of the set there exists such that . As in [Citation19], the zero-divisor graph of P is the graph G(P) whose vertices are the elements of such that two vertices a and b are adjacent if and only if .
We set . The elements are the dense elements of P. The zero-divisor graph of P with vertex set is and a and b are adjacent if and only if .
Throughout, P denotes a poset with 0 and qi, are all atoms of P, where .
Afkhami et al. [Citation3] partitioned the set as follows.
Let . The notation stands for the set:
In [Citation3], the following observations are proved.
Let and be the index sets of and , respectively. Then , if .
.
From the above two properties, it is clear that the sets forms a partition of . In fact, one can see that a relation on defined as follows: if and only if for some partition of is an equivalence relation.
The following discussion can be found in [Citation17].
The set of equivalence classes under of will be denoted by . Now, we set . We define relation on as follows. if and only if , for some and for some , where and for all . It is not very difficult to prove that is a poset. The least element of is P0, and if P has the greatest element 1, then the greatest element of the poset is . Sometimes, we abuse the notation and write as PI, where
Since is a poset with the least element , and except (where ), every element of is a zero-divisor. Note that may be empty. Hence the zero-divisor graphs and of the poset are same, that is, . Hence afterwards, we write for the zero-divisor graph of . Clearly, a and b are adjacent in G(P) if and only if and are adjacent in . More about the inter-relationship between the properties of G(P) and are mentioned in Lemma 1.17.
The following statements are essentially proved in [Citation17] and will be used frequently in the sequel.
Lemma 1.17.
[17, Lemma 2.11] The following statements are true.
If are distinct atoms of , then are distinct atoms of .
The elements in if and only if .
The elements and are adjacent in if and only if . Further, if and only if .
Let . Then for any in . Hence, the vertices of forms an independent set in . Further, if in , then for any and for any in . In particular, and are adjacent in with , then the vertices of and forms an induced complete bipartite subgraph of .
Theorem 1.18.
(Joshi [Citation15, Corollary 2.11]) Let G(P) be the zero-divisor graph of an atomic poset P. Then , where n is the finite number of atoms of P.
Theorem 1.19.
Let P be a finite poset with 0 and G(P) be its zero-divisor graph. Then Number of atoms of P.
Proof.
Let be the all n atoms of P. Hence, will also have exactly n atoms . Then . Since the zero-divisor graph is nothing but the and . Therefore, to prove , it is enough to prove that . Define a map by , where are distinct primes. We show that f is a coprime labeling of . By Lemma 1.17(3), PI and PJ are adjacent in if and only if if and only if and are coprime if and only if and are coprime. This proves f is a coprime labeling of and , i.e., , by Theorem 1.5. The result follows from Theorem 1.18. □
Remark 1.20.
Note that may not be true. Consider the Boolean lattice given in . It is easy to check that , however, .
Definition 1.21.
(West [Citation23]) The disjoint union of graphs is an operation that combines two or more graphs to form a larger graph. It is analogous to the disjoint union of sets and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs and by making the edge set of the result be the disjoint union of the edge sets of the given graphs. Any disjoint union of two or more nonempty graphs is necessarily disconnected. If G and H are two graphs, then denotes their disjoint union.
Remark 1.22.
It is easy to observe that for any graph G, , where Im is an independent set with m vertices. In particular, .
2 Applications
In this section, we show that various graphs associated with commutative rings with identity can be studied via zero-divisor graphs of specially constructed posets.
I. Zero-divisor graphs of finite reduced commutative rings
In [Citation10], it is observed that if R is a reduced Artinian ring with exactly k prime ideals, then there exist fields such that . Further, it is proved that the ring-theoretic zero-divisor graph equals the poset-theoretic zero-divisor graph of R (R treated as a poset under the partial order given in [10, Lemma 3.3]), that is, (cf. [Citation18, Remark 3.4]).
Corollary 2.1. Let R be a finite reduced commutative ring with identity and be its zero-divisor graph. Then Number of prime ideals of R.
II. Annihilating ideal graphs, Co-annihilating ideal graphs and Comaximal ideal graphs of commutative rings
Let R be a commutative ring with identity that is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists such that .
Definition 2.2.
(Visweswaran and Patel [Citation22]) Let R be a commutative ring with identity. Associate a simple undirected graph with R, called as the annihilating ideal graph of R, where the vertices of are nonzero annihilating ideals of R and two vertices I and J are adjacent if and only if I + J is also an annihilating ideal of R.
The following is the modified definition of an annihilating ideal graph. The modified annihilating ideal graph be a simple undirected graph with vertex set is set of all nonzero proper ideals of R and two distinct vertices I, J are joined if and only if I + J is also an annihilating ideal of R. By [Citation2, Lemma 2.1], I – J is an edge of if and only if .
Let Id(R) be the set of all ideals of R. Consider an equivalence relation on Id(R) such that if and only if . The equivalence class of an ideal I is the set , denoted by . The set of all equivalence classes of Id(R) is , denoted by . It is easy to observe that and , if .
Now, we define a relation on such that if and only if . It is relatively easy to prove that is a partial order on . Thus, is a poset. Since and , then is the and is the of .
We construct a poset from the poset . The elements and of the poset are denoted by (0) and R respectively, in the poset . For any element of , we replace the element of by the chain of elements of the equivalence class in whose length is equal to in some pre-determined well-order. Note that the element of is replaced by the one element chain, namely R, in irrespective of the cardinality of . Clearly, every element is an annihilating ideal of R. We call this poset the corresponding poset of the ring R.
We illustrate the above discussion with an example. Let , where F2 is the field of two elements. We can check that, . Also, . This gives, . The Hasse diagrams of the ideal lattice and the constructed poset are shown in .
From the construction mentioned above, the greatest element and the least element of a poset are the ideals (0) and R, respectively. Let . Clearly, I and J are comparable in if and only if either I and J in the same equivalence class, or and are comparable in .
Remark 2.3.
(1) We observe that is a meet-semi lattice. For this, let . Since , we have is a meet-semi lattice. Further, the poset is constructed from by replacing the elements of by the chains. Hence is a meet-semi lattice.
(2) Let R be an Artinian ring. Hence for every proper ideal I of R. Then the number of atoms of is equal to the number of maximal ideals of R. For this, we prove that has number of atoms given by . Let . Then . Hence and . Therefore, and . Also, for any for some . Therefore, . Hence . This proves are all distinct atoms of . Thus, by the construction of , the number of atoms of is equal to the number of maximal ideals of R.
The elements I and J are adjacent in if and only if for . By Remark 2.3, I and J are adjacent in if and only if . Note that, we can easily verify for the above example. With this preparation, we prove that the complement of an annihilating ideal graph of R is nothing but the zero-divisor graph of .
Lemma 2.4.
Let R be a commutative ring with identity that is not an integral domain, and let be the corresponding poset of R. Then .
Proof.
First we prove that the vertex set of and are equal. For this, let . Then I is an annihilating ideal of R, and I is an element of . Clearly, and . This gives that . Therefore . This proves .
Now, assume that . Clearly, . By the construction of , we have and . Thus and . This gives . As already observed is an annihilating ideal of R, . This proves . Thus, .
Now, we prove that I and J are adjacent in if and only if I and J is adjacent in . For this, suppose that I and J are adjacent in . By (1), I – J is an edge of if and only if . This implies , where R is the zero of the meet-semi lattice . Thus, I and J are adjacent in .
Assume that I and J are adjacent in . This implies . Thus, I and J are adjacent in . This proves . □
Definition 2.5.
(Akbari et al. [Citation4]) The co-annihilating-ideal graph of R, denoted by , is a graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann Ann .
Definition 2.6.
(Ye and Wu [Citation24]) Let R be a commutative ring with identity. We associate a simple undirected graph with R, called the comaximal ideal graph of R, where the vertices of are the proper ideals of R that are not contained in the Jacobson radical J(R) of R and two vertices I and J are adjacent if and only if I + J = R.
The following is the modified definition of the comaximal ideal graph. The modified comaximal ideal graph is the graph with the vertex set as the nonzero proper ideals of R and two vertices I and J are adjacent if and only if I and J are comaximal. Clearly, one can see that the comaximal ideal graph is the subgraph of the modified comaximal ideal graph . Moreover, , where .
Corollary 2.7.
[Citation4, Corollary 1.2] If R is an Artinian ring, then , where .
By Theorem 1.19, Remarks 1.22, 2.3(2), Lemma 2.4, Corollary 2.7, we have:
Corollary 2.8.
Let R be an Artinian ring with finitely many ideals. Then .
III. Intersection graphs of ideals of Artinian principal ideal rings
Let R be a commutative ring with identity. Then the intersection graph of ideals of R is the graph whose vertices are the nonzero proper ideals of R such that distinct vertices I and J are adjacent if and only if ; see [Citation9]. The vertex set of the zero-divisor graph of a lattice Id(R) is , that is, the set of nonzero proper ideals of R. Therefore . The ideals I and J are adjacent in if and only if , that is, the ideals I and J are adjacent in if and only if . The ideals I and J are adjacent in if and only if . Hence the following result.
Lemma 2.9.
Let R be a commutative ring with identity. Then .
The following discussion can be found in [Citation10].
Let R be a commutative ring with identity. Recall that R is a special principal ideal ring (SPIR for brevity) if R is a local Artinian principal ideal ring (cf. [Citation14]). If R is an SPIR with maximal ideal M, then there exists such that , and if I is an ideal of R, then for some ([14, Proposition 4]). In this case, M is nilpotent with an index of nilpotency equal to n, and the lattice of ideals of R is isomorphic to the chain C of length n (chain of n + 1 elements). By [14, Lemma 10], R is an Artinian principal ideal ring if and only if there exist SPIRs such that (it is also a straightforward consequence of the structure theorem of Artinian rings in [Citation6, Theorem 8.7]). Thus, Id(R) of an Artinian principal ideal ring is product of chains (Ci is chain of length ni), where , nilpotency index of maximal ideal Mi of Ri is ni. By Lemma 2.9 and from the above discussion, we have the following corollary.
Corollary 2.10.
Let R be an Artinian principal ideal ring. Then .
Acknowledgments
The authors are grateful to the referees for their suggestions which improved the presentation of the paper.
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References
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