The Metric Chromatic Number of Zero Divisor Graph of a Ring

Abstract

Let be a nontrivial connected graph, be a vertex colouring of , and be the colouring classes that resulted, where . A metric colour code for a vertex a of a graph is , where is the minimum distance between vertex and vertex in . If , for any adjacent vertices and of , then is called a metric colouring of as well as the smallest number satisfies this definition which is said to be the metric chromatic number of a graph and symbolized . In this work, we investigated a metric colouring of a graph and found the metric chromatic number of this graph, where is the zero-divisor graph of ring .

1. Introduction

It is supposed that is a commutative ring with identity, denoted the set of all nonzero zero divisors of a ring , and is an integer ring modulo . denotes a complete graph order while is a complete bipartite order . The cardinality of a set symbolizes |S|. If then we say that a divides , or is divisible by , and write and if a does divide , we write . If a ring has exactly one maximal ideal, then is called local. Obviously, is local if and only if for positive integer .

In [1], Anderson and Livingston define a zero-divisor graph of commutative ring and denoted this concept . This graph has vertices and edges ; they prove is connected, at most three, and is a finite graph if and only if is a finite ring. This concept is a mix between two branches of mathematical graph theory and ring theory. This idea comes from Beck in [2] when he studied the coloring of commutative rings. After that, Anderson and Livingston modified this concept. From this time, many authors studied this graph and gave properties, see [3–6]. As well as, there are other studies on the zero divisor graph of ring integer modulo n, for example, see [7–10]. Also, in graph theory, Chartrand et al. define a metric chromatic number of a graph and gave some basic properties [11]. Many authors studied this concept; see, for example, [12, 13].

This work aims to study a chromatic number of zero divisor graphs of the ring integer modulo . Section 2 gives some basic definitions and theorems of graph theory needed. In Section 3, we study a metric colouring of a graph and find all metric chromatic numbers of these graphs when for distinct prime numbers and positive integer numbers , and.

2. Definition and Terminologies

This section briefly discusses some definitions and theorems of the tile using graph theory.

Definition 1. (see [14]). A distance between two distinct vertices and in a graph denotes the length of the shortest path and a diameter of a graph is for all . While a distance between a vertex and a subset in is .

Remark 1. , and if and only if .

Definition 2. (see [14]). A vertex coloring of a graph is a function such that for every adjacent vertices and in and is called available colors. The smallest integer has a . This is called a chromatic number of and symbolized . Clearly, if has , then .

Definition 3. (see [14]). An induced subgraph of a graph that is complete is said to be a clique, and the number of the maximal clique is called the clique number and is denoted by .

Definition 4. (see [11]).Suppose that is a of a graph for a positive integer whenever adjacent vertices may be the same colour and let the colour classifications be . With each vertex , we can associate a and a symbolized called the metric colour code of , where for each with , . If for each pair of 's neighbouring vertices and , then is a metric colouring of . The smallest number , for which has a metric , is said to be the metric chromatic number of and is symbolized .
Clearly, is defined for every connected graph and for every nontrivial connected graph .

Remark 2. Let be a proper of a nontrivial connected graph with resulting colour classes and let two vertices and be joined by an edge in . Then, and for some with . Suppose that and . Then, and . Thus, and is also a metric colouring of . Thus, .

Lemma 1. (see Observation 1.2 in [11]). For a metric colouring of a connected graph and . If for each , then .

Remark 3. (see [11]). If is a complete graph order , then . So, if is a graph containing a complete subgraph with for all and , then .

3. Main Results

In this section, we look at a colouring code of a graph , and we find the metric chromatic number of this graph. We begin this section with local ring. Firstly, we study case .

Lemma 2. For every prime number , if , then .

Proof 1. By [15], therefore, Γ(R) has subgraph and graph so that .

Proposition 1. Let , where is a prime number; then, .

Proof 2. Since , so we show that . Assume, to the contrary, that . Then, there is is a of a graph . However, we can rewrite the vertices set of a graph into the two disjoint subsets as follows: , , and . If and , then , , , and for some positive integers , , and or do not divide p. Therefore, , , and . This implies every element in adjacent to every element in and as well as every two elements in are nonadjacent to each other. Since for all , with , where , then . Also, if , where without loss generality, say , then . However, adjacent with is a contradiction. Therefore, .

Example 1. Let ; then, . We can write , where andLet , and . Clearly, every two element x and y in are nonadjacent. Then, , and , for all .
The following result shows how we can extend Proposition 1 for every positive integer .

Theorem 1. Let , where p is a prime number and ; then,

Proof 3. We can classify as disjoint subsets , for all . When is an even number, obviously, , and if , then induced a complete subgraph of and , for all and . Therefore, by Lemma 1, , so Remark 3 leads to .
We claim that ; if not, then there are metric code of , say colours such that has colour . Now, for any vertex , c(y) has colouring , where . Since is adjacent to every vertex in , then which is a contradiction. Therefore, . Also, we can show has metric –colouring code of as follows: , where . Also, since and , then . Therefore, we can take every element in in one independent set , where j = 1, 2, …, , so that .
If is an odd number, it is proved in a similar way that is even; we have . Also, has metric as follows: , , and , where Therefore, .
Clearly, if , where and are distinct prime numbers, then is a complete bipartite graph. So, by [11], . Therefore, we find , whenever with and are distinct prime numbers and positive integer .

Theorem 2. For any two distinct prime numbers and , .

Proof 4. Let . By [8], we can rewrite , as disjoint subsets , and , where , , , and for positive integer less than and not divided or . We can note if and only if and or d or , for i = 1,2. Also,  =  -1.
First, has metric code shown as , , and then for all , ; also, if , then and if , then . It follows that . Second, if we assume that have metric code, since for all for all , then, by Lemma 1, . Therefore, every must be in different subset. So, we have at least subsets, and for all , we get . Let , without loss of generality; we assume that has the same colour of ; then, , but adjacent with which is a contradiction. Therefore, has metric and .
The next result affirms that we have extended Theorem 2, for a ring with distinct prime numbers and positive integer .

Theorem 3. If a ring isomorphic with , where are distinct prime numbers and positive integer numbers , then where

Proof 5. We can rewrite the set of all zero divisors of disjoint subsets as follows:
, and , where , , and and are positive integers less than with not divided to or and not divided to . Clearly, ; also, for any positive integer s, we get . Now, in a way that is easy, we can see every two elements in a subset are adjacent and , where and ; therefore, . Also, if is an odd number, then every is adjacent with y, where . So, by the same way of proof and through Theorem 2, we can show has code of a graph and , where

Example 2. Suppose that ; then, , whereSince , and , then and so that :
where,
We note , , and c, for all , when x∈ , , when x∈ , , when x∈ , , and when x∈ , for x∈ .

Theorem 4. If a ring isomorphic with , where are distinct prime numbers and positive integer numbers , then where

Proof 6. If we rewrite as and , i = 1, 2, …, , then we have an induced complete subgraph of the set and for all , . Furthermore, . So, by the same way of proof, through Theorems 2 and 3, we obtain

Data Availability

No data were used to support the findings of the study because this paper is in pure mathematics (algebra and graph theory) and all results are in these fields.

Conflicts of Interest

The authors declare that they have no conflicts of interest.