Abstract

The locating chromatic number of a graph is defined as the cardinality of a minimum resolving partition of the vertex set such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in are not contained in the same partition class. In this case, the coordinate of a vertex in is expressed in terms of the distances of to all partition classes. This concept is a special case of the graph partition dimension notion. In this paper we investigate the locating chromatic number for two families of barbell graphs.

1. Introduction

The partition dimension was introduced by Chartrand et al. [1] as the development of the concept of metric dimension. The application of metric dimension plays a role in robotic navigation [2], the optimization of threat detecting sensors [3], and chemical data classification [4]. The concept of locating chromatic number is a marriage between the partition dimension and coloring of a graph, first introduced by Chartrand et al in 2002 [5]. The locating chromatic number of a graph is a newly interesting topic to study because there is no general theorem for determining the locating chromatic number of any graph.

Let be a connected graph. We define the distance as the minimum length of path connecting vertices and in , denoted by . A -coloring of is a function , where for any two adjacent vertices and in . Thus, the coloring induces a partition of into color classes (independent sets) , where is the set of all vertices colored by the color for . The color code of a vertex in is defined as the -vector , where for . The -coloring of such that all vertices have different color codes is called a locating coloring of . The locating chromatic number of , denoted by , is the minimum such that has a locating coloring.

The following theorem is a basic theorem proved by Chartrand et al. [5]. The neighborhood of vertex in a connected graph , denoted by , is the set of vertices adjacent to .

Theorem 1 (see [5]). Let be a locating coloring in a connected graph . If and are distinct vertices of such that for all , then . In particular, if and are non-adjacent vertices of such that , then .

The following corollary gives the lower bound of the locating chromatic number for every connected graph .

Corollary 2 (see [5]). If is a connected graph and there is a vertex adjacent to leaves, then .

There are some interesting results related to the determination of the locating chromatic number of some graphs. The results are obtained by focusing on certain families of graphs. Chartrand et al. in [5] have determined all graphs of order with locating chromatic number , namely, a complete multipartite graph of vertices. Moreover, Chartrand et al. [6] have succeeded in constructing tree on vertices, , with locating chromatic numbers varying from to , except for . Then Behtoei and Omoomi [7] have obtained the locating chromatic number of the Kneser graphs. Recently, Asmiati et al. [8] obtained the locating chromatic number of the generalized Petersen graph for . Baskoro and Asmiati [9] have characterized all trees with locating chromatic number . In [10] all trees of order with locating chromatic number were characterized, for any integers and , where and . Asmiati et al. in [11] have succeeded in determining the locating chromatic number of homogeneous amalgamation of stars and their monotonicity properties and in [12] for firecracker graphs. Next, Wellyyanti et al. [13] determined the locating chromatic number for complete -ary trees.

The generalized Petersen graph , and , consists of an outer -cycle , a set of spokes , , and edges , , with indices taken modulo . The generalized Petersen graph was introduced by Watkins in [14]. Let us note that the generalized Petersen graph is a prism defined as Cartesian product of a cycle and a path .

Next theorems give the locating chromatic numbers for complete graph and generalized Petersen graph .

Theorem 3 (see [6]). For , the locating chromatic number of complete graph is .

Theorem 4 (see [8]). The locating chromatic number of generalized Petersen graph is for odd or for even .

The barbell graph is constructed by connecting two arbitrary connected graphs and by a bridge. In this paper, firstly we discuss the locating chromatic number for barbell graph for , where and are complete graphs on and vertices, respectively. Secondly, we determine the locating chromatic number of barbell graph for , where and are two isomorphic copies of the generalized Petersen graph .

2. Results and Discussion

Next theorem proves the exact value of the locating chromatic number for barbell graph .

Theorem 5. Let be a barbell graph for . Then the locating chromatic number of is .

Proof. Let , , be the barbell graph with the vertex set = and the edge set = ∪∪.
First, we determine the lower bound of the locating chromatic number for barbell graph for . Since the barbell graph contains two isomorphic copies of a complete graph , then with respect to Theorem 3 we have . Next, suppose that is a locating coloring using colors. It is easy to see that the barbell graph contains two vertices with the same color codes, which is a contradiction. Thus, we have that .
To show that is an upper bound for the locating chromatic number of barbell graph it suffices to prove the existence of an optimal locating coloring . For we construct the function in the following way: By using the coloring , we obtain the color codes of as follows: Since all vertices in have distinct color codes, then the coloring is desired locating coloring. Thus, .

Corollary 6. For , and , the locating chromatic number of barbell graph is

Next theorem provides the exact value of the locating chromatic number for barbell graph .

Theorem 7. Let be a barbell graph for . Then the locating chromatic number of is

Proof. Let , , be the barbell graph with the vertex set and the edge set = , ∪∪∪.
Let us distinguish two cases.
Case ( odd). According to Theorem 4 for odd we have To show that is an upper bound for the locating chromatic number of the barbell graph we describe an locating coloring using colors as follows: For odd the color codes of areSince all vertices in have distinct color codes, then the coloring with colors is an optimal locating coloring and it proves that .
Case ( even). In view of the lower bound from Theorem 7 it suffices to prove the existence of a locating coloring such that all vertices in have distinct color codes. For even, , we describe the locating coloring in the following way: In fact, our locating coloring of , even, has been chosen in such a way that the color codes areSince for even all vertices of have distinct color codes then our locating coloring has the required properties and . This concludes the proof.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are thankful to DRPM Dikti for the Fundamental Grant 2018.