Minimum Partition of an r − Independence System

Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), Bahauddin Zakariya University Multan, Multan 60800, Pakistan Department of Mathematics, &e Women University, Multan, Pakistan Department of Mathematics, &e Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan Department of Mathematics, Division of Science and Technology, University of Education Lahore, Lahore 32200, Pakistan Institute de Mathematiques de Jussieu-Paris Rive Gauche-Paris, Paris, France


Introduction
An abstract idea of representing any objects which are connected to each other in a form of relation is a graph. In this representation, the object is called as a vertex and their relation denotes as an edge. Partition of a graph is the distribution of the whole graph data into disjoint subsets to different devices. e need of distributing huge graph data set is to process data efficiently and faster process of any graph related applications. Where graph partitioning is essential and applicable are given as follows: (1) Complex networks which include biological networks (in solving biological interaction problem in a huge a biological network), social networks (Facebook, Twitter, and LinkedIn etc., and graph partitioning technology is used to process user query efficiently, as replying a query in a distributed manner is very handy and effective) [1], and transportation networks (graph partitioning can speed up and could be effective in planning a route by using a GPS (global positioning system) tool in the digital era). (2) PageRank, which is an application used to compute the rank of web rank from web network. (3) VLSI design: Very large-scale integration (VLSI) system is one of the graph partitioning problems in order to reduce the connection between circuits in designing VLSI. e main objective of this partitioning is to reduce the VLSI design complexity by splitting them into a smaller component.
(4) Image processing: Graph partitioning is one of the most attractive tools to split into several components of a picture, where pixels are denoted by vertices and if there are similarities between pixels are represented as edges [2].
Inspired by these interesting applications of graph partition, we consider a graph partition in the context of resolving set of a graph, which is a well-known parameter in graph theory and having remarkable application in network discovery and verification.
A set system is a finite set S together with a family F of subsets of S and is denoted by the pair (S, F). A set system (S, P) is said to be an independence system if for every subset X of S possessing property P, each proper subset of X also possesses the property P, i.e., for each X ⊂ S such that X ∈ P, Y ∈ P for all Y ⊂ X. Actually, in an independence system (S, P), P identified with the family of subsets of S possessing the property P. A subset X of S which possess the property P is said to be an independent set and dependent set otherwise. e chromatic number of (S, P) is the smallest natural number n such that S can be partitioned into n independent sets and is denoted by χ(S, P). Clearly, a partition of S into n independent sets of (S, P) can be identified by a coloring λ: S ⟶ 1, 2, . . . , n { } of S such that for each color c ∈ 1, 2, . . . , n { }, the color class s ∈ S, λ(s) � c { } has the property P, and vice versa. e coloring λ of S is called a P−coloring of S. us, χ(S, P) is the least number of colors required by a P−coloring of S and is also called the P−chromatic number of S [3]. e P−chromatic number χ(S, P) has been extensively studied by various graph theorists. Remarkable work has been done when S is V or E for a graph G having vertex set V and edge set E, and P is a hereditary graphical property. For example, if P is the property I of being a vertex independent set, then χ(V, I) is the ordinary chromatic number of G; if P is the property E of being an edge independent set, then χ(E, E) is the edge chromatic number of G; if P is the property F of being a forest, then χ(E, F) is the arboricity of G. In the next section, we consider P as the property R of being a resolving set for G and define the R−chromatic number of G associated with an r−independence system (V, R).

r − Independence System
Hereafter, we consider nontrivial, simple, and connected graph G with vertex set V and edge set E. We denote two adjacent vertices u and v in G by u ∼ v and nonadjacent vertices by u≁v. e distance d: { } is the length of a shortest path between two vertices in the pair (u, v) ∈ V × V and is denoted by d (u, v). e maximum distance between the vertices of G is called the diameter of G, denoted by diam(G). Two vertices u and v in G are antipodal or diametral if d(u, v) � diam(G); otherwise, they are nonantipodal.
Let G be a graph. For any vertex v of G, the metric code or code of v with respect to an ordered k−subset W � w 1 , w 2 , . . . , w k of V is defined as An ordered k−subset W of V is a resolving set for G if c W (u) ≠ c W (v) for every pair of vertices (u, v) ∈ V × V. e cardinality of a minimum resolving set for G is called the metric dimension of G, denoted by dim(G) or β(G). A resolving set for G of cardinality dim(G) is called a metric basis or a basis of G [4][5][6][7][8]. In [9], it was found and, in [6], an explicit construction was given that finding the metric dimension of a graph is NP-hard. e concept of a resolving set, other than graph theory, is applied in many other areas such as coin-weighing problems [10], network discovery and verification [2], strategies for mastermind games [11], pharmaceutical chemistry [12], robot navigation [6], connected joins in graphs and combinatorial optimization [13], and sonar and coast guard Loran [8].
A subset S of the vertex set V of a graph G is an r−independent set if no proper subset of S is a resolving set for G. We denote R as the property of being an r−independent set. at is, a subset S of V possesses the property R if and only if S is an r−independent set. is concept was firstly introduced by Boutin and used the term res-independent set [14]. For simplicity, we use the term r−independent set rather than res-independent set. A family of subsets of V possessing the property R is defined as (2) us, we have a set system (V, R) consisting of those subsets of V which are possessing the property R and is called the r−independence system. All the subsets possessing the property R may or may not be resolving. is was an error made by Boutin in [14], and we rectified it in [15]. en, resolving set for G (3) a minimal resolving set for G is a maximal r−independent set, but converse is not always true [15] 2.1. Minimum Partition Problem. For a connected graph G � (V, E) and the r-independence system (V, R), the minimum partition problem is to make a partition of V into the minimum number of subsets possessing the property R. e least natural number k, such that V can be partitioned into k subsets possessing the property R, is called the resolving chromatic number of G associated with (V, R), denoted by χ r (V, R). A coloring λ: us, χ r (V, R) is the least number of colors required by an R−coloring of V and is also called the R−chromatic number of G. 1 . en only two colors are needed to properly color V, and it follows that the ordinary chromatic number χ(G) � 2 with color classes v 1 , v 3 and v 2 , v 4 . e metric dimension of G is 2 and two nonantipodal vertices of G form a basis of G [4]. Accordingly, no 3−element subset of V possess the property R and so the r−independence system is e minimum partition of V according to R−coloring of V consists of two 2−element subsets of V from (V, R), and hence, χ r (V, R) � 2.
In the above example, we obtained that the chromatic number and R−chromatic number of a graph G are same. But, it is not necessary that these numbers are always same.
en only two colors are needed to properly color V, and it follows that the ordinary chromatic number χ(G) � 2 with color classes v 1 , v 3 and v 2 , v 4 . e metric dimension of G is 1, and v 1 , v 4 are the only two bases of G [4,6]. Accordingly, each 2−element subset of V is a resolving set for G, and so no 3−element subset of V possess the property R.
us, the r−independence system is e minimum partition of V according to R−coloring of V consists of two bases sets and the set v 2 , v 3 from (V, R).
It is observed, from Examples 1 and 2, that χ(G) ≤ χ r (V, R). But, it is not true generally as, in the next example, we have a have a graph G such that en, three colors are required to properly color V, and it follows that the ordinary chromatic number χ(G) � 3 with color classes v 1 , v 2 and v 3 . e metric dimension of G is 2 and any two vertices of G can form a basis of G [4]. Accordingly, the 3−element set V does not possess the property R and so the r−independence system is 5 . en two colors are required to properly color V, and it follows that the ordinary chromatic number χ(G) � 2 with color classes v 2 , v 4 , v 5 and v 1 , v 3 . e metric dimension of G is 2 and v 4 , v 5 is a set of basis of G [4]. But the set of three elements v 1 , v 2 , v 3 which is not resolving set of G possess the property R and so the r−independence system is e minimum partition of V according to R−coloring of V consists of one 2−element set and one

Three Well-Known Families
In this section, we consider families of path graphs, cycle graphs, and wheel graphs and solve the minimum partition problem for each family.

Path Graphs.
A path graph P n , for n ≥ 2, has vertex set e following result describes which subset of the vertex set of a path graph possesses the property R.
Proof. According to Lemma 1, each singleton subset of V as well as each 2−element subset of V − v 1 , v n possesses the property R. It follows that for n � 2, 3, (V, R) � v i ; 1 ≤ i ≤ n , and for all n > 3, Journal of Mathematics where V 2 denotes the collection of all the n − 2 2 and 2−element subsets of V − v 1 , v n . Hence, □ e following result solves the minimum partition problem for a path graph. Proof. Let the color classes, due to a coloring λ: When n is even, then In both cases, all these color classes are lying in (V, R), by Lemma 2. It follows that λ is an R−coloring of V, and these color classes define a partition of V into the sets possessing the property R. Further, any partition of V of cardinality less than ⌊(n + 3)/2⌋ contains at least one us, a minimum partition of V has ⌊(n + 3)/2⌋ subsets of V possessing the property R.
In the next result, we investigate which subset of the vertex set of a cycle graph possesses the property R is.

Lemma 3.
A subset of the vertex set V of a cycle graph G which possess the property R is a singleton set or a 2−element set.
Proof. In [4,5], it was shown that dim(G) � 2 and any two nonantipodal vertices of G form a basis of G. Further note that, a 3−element subset S of V, whether containing two antipodal vertices or not, is not an r−independent set. It completes the proof. □ e number of subsets of the vertex set of a cycle graph possessing the property R is counted in the following result.
Proof. Lemma 3 yields that each singleton subset of V as well as each 2−element subset of V possesses the property R. It follows that for all n ≥ 3, where V 2 denotes the collection of all the n 2 , 2−element subsets of V. Hence, □ e minimum partition problem for a cycle graph is solved in the following result. Proof. Let the color classes, due to a coloring λ: V ⟶ 1, 2, . . . , ⌈n/2⌉ { } of V, are as follows: When n is even, then In both the cases, all these color classes are lying in (V, R), by Lemma 4. It follows that λ is an R−coloring of V, and these color classes define a partition of V into the sets possessing the property R. Further, any partition of V of cardinality less than ⌈n/2⌉ contains at least one 3−element subset S of V such that S∈(V, R). us, a minimum partition of V has ⌈n/2⌉ subsets of V possessing the property R. □ 3.3. Wheel Graphs. For n ≥ 3, let C n : v 1 ∼ v 2 ∼ · · · ∼ v n ∼ v 1 be a cycle and K 1 be the trivial graph with vertex v. en a wheel graph is the sum For fixed i; 1 ≤ i ≤ n, let a path P: v i ∼ v i+1 ∼ · · · ∼ v i+n−5 of order n − 4 on the cycle C n of W n , where the indices greater than n or less than zero will be take modulo n. e following result describes the sets in W n possessing the property R for n ≥ 8.

Remark 3.
For n ≥ 8, let W n be a wheel graph with vertex set u. Let V be set of any v i , i � 1, . . . , n − 4 consecutive vertices of wheel graph, then V is a maximal independent set which is not a minimal resolving set.

Lemma 5.
For n ≥ 8, let W n be a wheel graph with vertex set V. en, for any W⊆S. It follows the required result. ( is implies that c W (x) � c W (y) for any W ⊆ V − S. Hence, the required result followed.

□
For wheel graphs, the following result solves the minimum partition problem.
Proof. It can be easily seen that a partition 6 is a minimum partition of V having sets possessing the property R in W 6 , and 7 , v is a minimum partition of V having sets possessing the property R in W 7 .
For all n ≥ 8, let λ: V ⟶ 1, 2 { } be a coloring of V and let the corresponding color classes are C 1 � V(P) ∪ v { }, where P: v i ∼ v i+1 ∼ · · · ∼ v i+n−5 for any fixed 1 ≤ i ≤ n, and C 2 � V − C 1 . en C 1 and C 2 define a partition of V. Also, Lemma 5 yields that both C 1 and C 2 possess the property R. erefore, λ is an R−coloring of V, and hence χ r (V, R) � 2.  Either u, v are adjacent or nonadjacent twins, they are called

Twins and r − Independence
, for all u ∈ V. Each self twin in a graph makes a set of singleton twins. A set T⊆V is called a twin set in G if u, v are twins in G for every pair of distinct vertices u, v ∈ T. e next lemma follows from the above definitions [16].
Lemma 6 (see [16]). If u and v are twins in a graph G, then Due to Lemma 6, we have the following remark.

Remark 4
(1) If u and v are twins in a graph G and W is a resolving set for G. en either u ∈ W or v ∈ W. (2) If T is a twin set in a graph G of order t ≥ 2, then every resolving set for G contains at least t − 1 elements of T.
Removal of two twins from the vertex set makes it r−independent as given in the next result.

Lemma 7. Let T be a twin set of order t ≥ 2 in a graph G having the vertex set V. en, for any two elements
{ } is a resolving set for G. It follows the result. □ Remarks 2 and 4 yield the following two results.

Lemma 8. Let T be a twin set of order t ≥ 2 in a graph G.
en, for each 1 ≤ k ≤ t − 1, any k−element subset S of T and the set T − S both possess the property R.
Proof. If t � 2, then each subset S of T of order less that |T| as well as the set T − S both are singleton, and so Remark 2(1) yields the result. If t ≥ 3, then as t − 1 vertices of T must belong to any resolving set for G, by Remark 4(2), so no subset of T of order ≤t − 2 is not a resolving set, because there are at least two twins are remained in T form that one of them must belong to a resolving set for G, by Remark 4(1). It follows that any k−element subset S of T possesses the property R for each 1 ≤ k ≤ t − 1. Further, since the set T − S is either singleton or contains at least more than one twins from T, no subset of T − S is a resolving set for G. us, it must possess the property R. □ Lemma 8 can be generalized with the similar proof when a graph G has more than one twin sets of order at least two, and this generalization is stated in the following result.

Journal of Mathematics
Remark 5. Let G be a family of graph and G ∈ G. For l ≥ 2, let T 1 , T 2 , . . . , T l are twin sets in a graph G of orders t 1 , t 2 , . . . , t l , respectively, where each t i ≥ 2. Let a nonempty set S is union of singleton twin sets in G, and let v 1 , v 2 belong to any one of T 1 , T 2 , . . . , T l , then S ∪ T i − v 1 , v 2 is a maximal independent set which is not minimal resolving set. e following result states the relationship between twins and r−independence.

Theorem 5.
e R−chromatic number of a graph G (except P 3 ) of order n ≥ 3 having a nonsingleton twin set is two.
Proof. Let G be a graph of order n ≥ 4 with vertex set V, and let T be a twin set in G. Let λ: V ⟶ 1, 2 { } be a coloring of V, and let the corresponding color classes are en C 1 and C 2 define a partition of V. Also, Lemma 7 implies that C 1 ∈ (V, R). Further, since no path graph of order more than three has a twin set, so G is not a path graph. It follows that no singleton subset of C 2 is resolving, because a path graph only has a singleton resolving set [4]. us, C 2 ∈ (V, R). Hence, λ is an R−coloring of V, and so χ r (V, R) � 2.

Remark 6
(1) e converse of eorem 5 is not true generally. eorem 3 describes that the R−chromatic number of a wheel graph W n is two, but W n has no twin class for any n ≥ 8.
(2) In eorem 5, if G is P 3 , then G has one twin set containing two end vertices. But, the R−chromatic number of G is three, by eorem 1.
Next, we provide two well-known families of graphs as in the favor of eorem 5.
(i) If m i � 1 for all 1 ≤ i ≤ k, then G is a complete graph K k having vertex as the twin set. (ii) If some of m i is not equal to one. Let us suppose, without loss of generality, that m i � 1 for 1 ≤ i ≤ l and 2 ≤ l < k. en, G has k − l + 1 twin sets As, the R−chromatic number of P 3 is 3, by eorem 1, so we receive the following consequence from eorem 5.

Corollary 1.
e R−chromatic number of a complete multipartite graph (which is not K 1,2 ) is two.
Example 5 (Circulant networks). e family of circulant networks is an important family of graphs, which is useful in the design of local area networks [17]. ese networks are the special case of Cayley graphs Cay(G; S) when the group G is Z n (an additive group of integers modulo n) and S ⊆ Z n \ 0 { } [18]. ese graphs are defined as follows: let n, m and a 1 , a 2 , . . . , a m be positive integers, 1 ≤ a i ≤ ⌊n/2⌋ and a i ≠ a j for all 1 ≤ i < j ≤ m. An undirected graph with the set of vertices v i+1 ; i ∈ Z n , and the set of edges v j ∼ v j+a l : 1 ≤ j ≤ n, 1 ≤ l ≤ m} is called a circulant graph, denoted by C n (a 1 , a 2 , . . . , a m ). e numbers a 1 , a 2 , . . . , a m are called the generators, and we say that the edge v j ∼ v j+a l is of type a l . e indices after n will be taken modulo n. C n (a 1 , a 2 , . . . , a m ) is called the principal cycle. Consider a class of circulant networks C 2n+2 (1, n), for n ≥ 1.
en there are n + 1 twin sets us, as a consequence of eorem 5, the R−chromatic number of C 2n+2 (1, n) is two.

Some Realizations
Remark 2(3) describes that there is no n−element r−independent set in a connected graph of order n ≥ 2. Lemma 7 illustrates that a connected graph G of order n ≥ 3 having twins (other than self twins) can have an (n − 2)−element set as an r−independent set. In the result to follow, we characterize all the connected graphs of order n ≥ 2 in which every (n − 1)−element subset of the vertex set is r−independent. Proof. Suppose that G is a complete graph. en every two vertices of G are twins and V itself is the twin set in G. So Lemma 8 yields the required result.
Conversely, suppose that any (n − 1)−element subset, say S � s 1 , s 1 , . . . , s n−1 , of V possesses the property R. en S is a resolving set for G, because for any s i ∈ S, 0 lies at the ith position in the code c S (s i ), whereas the code c S (v) of the element v ∈ V − S has all nonzero coordinates. Further, S is a minimum resolving set for G, because no k−element subset of S is a resolving set for any 1 ≤ k ≤ n − 2, by our supposition. us, dim(G) � n − 1. In [4,6], it was shown that a graph G of order n has dim(G) � n − 1 if and only if G is a complete graph. It completes the proof.

□
Since any singleton set is an r−independent, so we have the following consequences for a complete graphs.

Corollary 2.
e R−chromatic number of a complete graph is two.
If G is a connected graph of order n ≥ 2 with vertex set V, then 2 ≤ χ r (V, (R)) ≤ n, by Remark 2(3). e next result characterizes all the connected graphs of order n ≥ 2 having R−chromatic number n. Theorem 7. Let G be a connected graph of order n ≥ 2 with vertex set V. en, χ r (V, R) � n if and only if G is either K 2 (� P 2 ) or P 3 (� K 1,2 ).
If G � P 3 (� K 1,2 ), then χ r (V, R) � 3, by eorem 1. Conversely, suppose that for a connected graph G of order n ≥ 2 with vertex set V, we have χ r (V, R) � n. en, (i) for n � 2, the only connected graph is K 2 (� P 2 ) such that χ r (V, R) � 2 � n, by Corollary 2, (ii) for n � 3, G is either C 3 (� K 3 ) or P 3 (� K 1,2 ). Since the R−chromatic number of C 3 is 2 � n − 1, by eorem 2, and the R−chromatic number of P 3 is 3 � n, by eorem 1, so G � P 3 in this case. (ii) for n ≥ 4, either G has a twin set or no twin set exists in G. In the former case, χ r (V, R) � 2 ≠ n, by eorem 5. In the latter case, let V � v 1 , v 2 , . . . , v n . en, χ r (V, R) � n implies that the minimum partition of the r−independence system (V, R) is v 1 , v 2 , . . . , v n . It follows that no k−element subset of V belongs to (V, R) for k ≥ 2. Otherwise, χ r (V, R) ≤ n − 1. But, in every connected graph of order n ≥ 4, at least one 2−element subset of V must possess the property R, because singleton resolving sets exist in a path graph only (and in the case of path graph P n , (n ≥ 4), we have 2−element subsets of V in (V, R), by Lemma 2). erefore, no connected graph G of order n ≥ 4 exists such that χ r (V, R) � n.
From the above three cases, we conclude that G is either K 2 (� P 2 ) or P 3 (� K 1,2 ). □ From eorem 7, it concludes that if G is a connected graph of order n ≥ 3 with vertex set V and G � P 3 (� K 1,2 ), then 2 ≤ χ r (V, R) ≤ n − 1. All the connected graphs of order n ≥ 3 having R−chromatic number n − 1 are characterized in the following result. Proof. If G � C 3 (� K 3 ), then eorem 2 yields that χ r (V, R) � n − 1.
χ r (V, R) � 2 ≠ n − 1 in the latter case, by eorem 5. In the former case, except G � P n , a minimum resolving set for G is of at least two order, which implies that every 2−element subset of V belongs to (V, R). It follows that a minimum partition of V according to the r−independence system contains at least two 2−element subsets of V, which implies that χ r (V, R) ≤ n − 2, a contradiction. However, when G � P n , then χ r (V, R) � 4 � n − 1 only for n � 5 and χ r (V, R) ≤ n − 2 for n ≥ 6, by eorem 1.
From the above three case, we conclude that G is either C 3 (� K 3 ) or P 4 or P 5 .
□ eorem 8 concludes that if G is a connected graph of order n ≥ 4 with vertex set V and G � P 4 , P 5 , then 2 ≤ χ r (V, R) ≤ n − 2.

Metric Dimension and r − Independence
In this section, we develop a relationship between the metric dimension and R−chromatic number of a connected graph by providing three existing type results.
ere exists a connected graph whose metric dimension is different from its R−chromatic number by one.

Theorem 9. For even n ≥ 4, there exists a connected graph
Proof. Let C n be a cycle graph on even n ≥ 4 vertices and a path P 2 . en G is a graph obtained by taking the product of C n and P 2 . Let the vertex set of G be V � v i , u i ; 1 ≤ i ≤ n , and the edge set is e resultant graph G consists of two n cycles: one is outer cycle v 1 ∼ v 2 ∼ · · · ∼ v n ∼ v 1 , and the other one is inner cycle u 1 ∼ u 2 ∼ · · · ∼ u n ∼ u 1 . It is shown, in [19], that dim(G) � 3. Next, we investigate the R−chromatic number of G with the help of the following five claims: Claim 1. Every singleton, 2−element and 3−element subset of V possesses the property R. Based on Remark 2(1) and due to dim(G) � 3 [19], this claim is true, because a minimum resolving set for G is of cardinality 3, and so no singleton and 2−element subset of V is a resolving set for G. 2) , u i , u i+(n/2) ⊂ V possesses the property R. By Claim 2, no subset of Z is a resolving set for G. It follows the required claim. Claim 4. No r−independent set (other than Z) in G of cardinality greater than 3 contains any of the pairs (v i , v i+(n/2) ) and (u i , u i+(n/2) ). Let S be an r−independent set in G of cardinality greater than 3. Suppose, without loss of generality, S contains the pair (v i , v i+(n/2) ). en a subset v i , v i+(n/2) , v of S for v ∈ S − v i , v i+(n/2) is a resolving set for G, by Claim 2, which contradicts the r−independence of S. us, S cannot contain the pair (v i , v i+(n/2) ). Similarly, the pair (u i , u i+(n/2) ) will not contained in S. Claim 5. For each 1 ≤ k ≤ n, there is a k−element subset of V possessing the property R. Claim 1 yields the result of k � 1, 2, 3. For k � 4, we have a set in (V, R), by Claim 3. Next, keeping Claim 4 in mind, let us consider two subset of V of cardinality n as follows: for fixed 1 ≤ i ≤ n, where the indices greater than n or less than or equal to zero will be taken modulo n. en d(u i , x) � d(v i+1 , x) for all x ∈ S 1i and d(v i , y) � d(u i+1 , y) for all y ∈ S 2i . It follows that c W (u i ) � c W (v i+1 ) for any W ⊆ S 1i and c W (v i ) � c W (u i+1 ) for any W ⊆ S 2i . Hence, both the S 1i , S 2i and each subset of any cardinality all are the subsets of V possessing the property R, and of course, they are k−element subsets of V for 1 ≤ k ≤ n. Now, let λ: V ⟶ 1, 2 { } be a coloring of V, and let the corresponding color classes are, for fixed 1 ≤ i ≤ n, C 1 � u j , v l ; j � i + 1, i + 2, . . . , i + n 2 ∧ l � i, i − 1, . . . , i − n 2 + 1 , where the indices greater than n or less than or equal to zero will be taken modulo n. en C 1 and C 2 define a partition of V. Also, Claim 5 yields that C 1 , C 2 ∈ (V, R). Hence, λ is an R−coloring of V, and so χ r (V, R) � 2. erefore, dim(G) − χ r (V, R) � 1 for every even value of n ≥ 4. □ Remark 7. It is not necessary that the difference dim(G) − χ r (V, R) is constant (fixed) always. It can arbitrarily large depending upon the order of the graph. For instance, let G be a wheel graph of order n ≥ 8, then dim(G) � ⌊(2/5)(n + 1)⌋ [20], and χ r (V, R) � 2, by eorem 3. So, it can be seen that the difference dim(G) − χ r (V, R) � ⌊(2/5)(n − 4)⌋, which is depending upon n and is not fixed. e next result shows that there exists a connected graph whose R−chromatic number is different from its metric dimension by one.
Theorem 10. For odd n ≥ 3, there exists a connected graph G with vertex set V such that χ r (V, R) − dim(G) � 1.