The Locating Chromatic Number of Book Graph

Let G � (V(G), E(G)) be a connected graph and c: V(G)⟶ 1, 2, . . . , k { } be a proper k-coloring of G. Let Π be a partition of vertices ofG induced by the coloring c. We define the color code cΠ(v) of a vertex v ∈ V(G) as an ordered k-tuple that contains the distance between each partition to the vertex v. If distinct vertices have distinct color code, then c is called a locating k-coloring of G. (e locating chromatic number of G is the smallest k such that G has a locating k-coloring. In this paper, we determine the locating chromatic number of book graph.


Introduction
All graphs considered in this paper are assumed to be simple, connected, and undirected. Let G � (V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). By P n and S n , we denote a path on n vertices and a star on n + 1 vertices, respectively. e distance, d (u, v), between two vertices u and v is defined as the length of the shortest path connecting them in G.
For a graph G and a positive integer k, a coloring c: V(G) ⟶ 1, 2, . . . , k { } with c(u) ≠ c(v) for every two adjacent vertices u and v is called a proper k-coloring of G. Let Π � C 1 , C 2 , . . . , C k be a partition of vertices of G induced by the coloring c, where C i denotes the set of vertices where d(u, C i ) � min d(u, x): x ∈ C i for every i � 1, 2, . . . , k. If distinct vertices have distinct color codes, then c is called a locating k-coloring of G. e locating chromatic number of G, denoted by χ L (G), is the smallest k for which G has a locating k-coloring.
e concept of locating chromatic number of a graph was introduced in 2002 by Chatrand et al. [1]. is concept is derived from the graph partition dimension and graph coloring. e study about locating chromatic number of a graph has grown constantly. Chartrand et al. [1] determined the locating chromatic number of some well-known graph classes such as paths with χ L (P n ) � 3 for n ≥ 3, cycles with χ L (C n ) � 3 for odd n ≥ 3 and χ L (C n ) � 4 for even n ≥ 4, and double stars with χ L (S m,n ) � n + 1 for 1 ≤ m ≤ n and n ≥ 2. In [2], Behtoei and Omoomi investigated the locating chromatic number for Cartesian product of graphs. Welyyanti et al. [3] discussed the locating chromatic number for disconnected graphs. Furthermore, the locating chromatic number of spinner graph was proved by Rianti and Narwen [4].
Let G and H be two given graphs. e Cartesian product of G and H, denoted by G × H, is a graph having the vertex For a positive integer n ≥ 1, a book graph, denoted by B n , is a graph obtained from n copies of a cycle of order 4 sharing one common edge. e book graph B n is also called the Cartesian product P 2 × S n . In this paper, we investigate the locating chromatic number of book graph. e following theorem proved by Chartrand et al. [1] is useful in determining the locating chromatic number of a graph.

Result and Discussion
In this section, we determine the exact value for the locating chromatic number of book graphs. Let B n be a book graph with the vertex set V(B n ) � u i , v i : i � 0, 1, . . . , n and the edge set E(B n ) � u i v i : i � 0, 1, . . . , n ∪ u 0 u i , v 0 v i : i � 1, 2, . . . , n}.
Two observations below follow from definition of proper coloring.

Lemma 1. Let c be a locating coloring of the book graph
Proof. Start coloring the graph by giving color C 1 and C 2 for the vertices in the middle, that is, c(u 0 ) � C 1 and c(v 0 ) � C 2 . From Observation 2, we know that we should have c(u i ) ≠ C 1 and c(v i ) ≠ C 2 for each i � 1, 2, . . . , n. Case 2. If c(u a ) � 2, the total possibility of the colors being used is (k − 1) because c(v a ) can only be given with 1, 3, 4, . . . , k, where only one color can be used by book graph B n for n ≥ 2. e illustration is shown in Figure 1.

Case 3.
If c(v a ) � 1, the total colors can be used are (k − 1) because c(u a ) can only be given by color 2, 3, 4, . . . , k, where only two colors can be used on book graph B n for n ≥ 2. e illustration is shown in Figure 2.
Since the coloring c(v a ) � 1 and c(u a ) � 2 have been used in Case 2, then the coloring is not reusable, and thus, the total coloring is (k − 1) − 1. If uses k colors, then χ L (B n ) must satisfy the following (2)

Theorem 2. Locating chromatic number of book graph B n is
for k ≥ 4.
Proof. Firstly, we will prove the existence of χ L (B n ). We can clearly see that k: n ≤ 2 k − 2 2 + 2(k − 1) − 1 is the subset from a set of real numbers. So from the well-ordering principle, the set must have the smallest element, say k * :� min k: n ≤ 2 k − 2 2