ON INCLUSIVE LOCAL IRREGULARITY VERTEX COLORING OF BOOK GRAPH FAMILY

Article History: Let 𝐺(𝑉,𝐸) is a simple and connected graph with 𝑉(𝐺) as the vertex set and 𝐸(𝐺) as the edge set. Vertex labeling on inclusive local irregularity vertex coloring is defined by mapping 𝑙: 𝑉(𝐺) → {1,2,… ,𝑘} and the function of the inclusive local irregularity vertex coloring is 𝑤 𝑖 :𝑉(𝐺) → 𝑁 with 𝑤 𝑖 = 𝑙(𝑣) + ∑ 𝑙(𝑢) 𝑢∈𝑁(𝑢) . In other words, an inclusive local irregularity vertex coloring is defined by coloring the graph so that its weight value is obtained by adding up the labels of the neighboring vertices and its label. The inclusive local irregularity chromatic number is defined as the minimum number of colors obtained from coloring the vertex of the inclusive local irregularity in graph G, denoted by 𝜒 𝑙𝑖𝑠𝑖 (𝐺) . In this paper, we learn about the inclusive local irregularity vertex coloring and determine the chromatic number on the book graphs family.


INTRODUCTION
Graph theory is one of the discrete mathematics branches widely applied in various fields. A graph is defined as a pair of sets which are the infinite and non-empty sets of elements called vertices, whereas they are the edge set formed from the connecting line segments of two vertices [1]. There are several discussions in graph theory such as graph coloring. Graph coloring can be interpreted as giving a different color to each element of the graph, namely a vertex, edge, or neighboring region [2]. In this study, we used vertex coloring, thus the provision of color to all vertices on the graph so that every two neighboring vertices have a different color [2]. Vertex coloring on the graph is the giving of a different color to each vertex to a minimum so that the neighboring vertex has a different color [3]. The fundamental problem in the coloring of graph is to determine the minimum possible color needed to color any graph, which is then called the chromatic number [4]. The minimum positive integer k for graph G having k-node coloring is called the chromatic number G, and is annotated ( ) [5]. A vertex coloring on a graph is said to be efficient if the number of colors used to color a graph is the chromatic number of the graph [6].
There are several special cases in the vertex coloring of a graph, one of them is the inclusive local irregularity vertex coloring. An inclusive local irregularity vertex coloring is a development of the local irregularity vertex coloring concept. Irregularity means that the labeling of vertices can be repeated and local means that the weight of neighboring vertices must have different values. While inclusive itself comes from the word include, and in this study, it is interpreted by adding the value of the vertex label itself. At the coloring of the vertex of local irregularity, the weight of the vertex is obtained by summing the value of the neighboring vertex [7]. Meanwhile, the weight of the vertex on the coloring of the inclusive local irregularity vertex is obtained by summing the value of the vertex adjacent to the value of the vertex itself.
The research related to the inclusive local irregularity vertex coloring started from the research on several simple graphs, which include path graphs, cycle graphs, and star graphs [8]. The research became the initial research on the inclusive local irregularity vertex coloring, which was then continued by other studies, such as inclusive local irregularity vertex coloring in tree graph families, which is studied by [9], inclusive local irregularity vertex coloring on fan graphs, firecracker graphs, and sun graphs by [10] and inclusive local irregularity vertex coloring on unicyclic graph families and star graph comb operations which is studied by [11] [12].
Graph operation is a way to create a new graph which differs from the original graph [13]. One of the examples of graph operation is edge amalgamation. Edge amalgamation on graphs 1 and 2 is the fusion of the edge ∈ ( 1 ) and edge ∈ ( 2 ), denoted by ( 1 , 2 ; , ) [14]. Book graphs can be interpreted as graphs resulting from edge amalgamation on cycle graphs [7]. In this study, the book graph is denoted by where is the number of vertices on the cycle graph and n is the number of cycle graphs. The family of book graphs used in this study are book graphs ( 4 ), book graphs ( 5 ), and book graphs ( 6 ).

RESEARCH METHODS
This research uses the axiomatic deductive method and pattern detection method. The axiomatic deductive method is defined as a research method that uses a theorem or axiom with the principles of deductive proof in mathematical logic to then be used in local irregularity vertex coloring on the book graph. While the pattern detection method in this study is used to formulate patterns and look for inclusive local irregularity chromatic numbers in a book graph. We used several definitions, lemmas, and observations in order to prove the theorem. The definitions, lemmas, and observations are as follows: ∈ ( ), ( ) ≠ ( ).

Definition 2. [15] The minimum number of colors obtained by local irregularity vertex coloring of a graph G is called local irregularity vertex chromatic number, denoted by ( ).
Inclusive local irregularity vsertex coloring [8] and has obtained a definition of inclusive local irregularity vertex coloring as follows: . Labeling is an inclusive local irregularity vertex coloring which fulfill the following condition: i.

Definition 4. [8] The minimum number of colors obtained by inclusive local irregularity vertex coloring of a graph G is called inclusive local irregularity vertex chromatic number, denoted by ( ).
To make it easier to find an inclusive local irregularity vertex coloring in this study, the following lemma and observations will be used:

RESULTS AND DISCUSSION
In this article, we discuss the chromatic numbers of inclusive local irregularities in the book graph family, including book graphs ( 4 ), book graphs ( 5 ), and book graphs ( 6 ). The results about the inclusive local irregularity vertex chromatic number of the book graph family decompose in some theorems as follows: Theorem 1. Given a book graph ( 4 ) with ≥ 3, then the inclusive local irregularity chromatic number of the graph is ( 4 ) = 4.
Next, it will be investigated that the chromatic numbers of inclusive local irregularities on the book graph ( 4 ) with ≥ 3 is ( 4 ) = 4. Based on Lemma 1 which says that "For graphs G which are simple and connected graphs, then ( ) ≥ ( )" is obtained the lower bound of chromatic numbers of inclusive local irregularity from the graph book is ( ) ≥ ( ) = 4 with ≥ 3. Then to prove the value ( 4 ), will be found the upper bound of the chromatic number of inclusive local irregularities through the function of labeling the vertex defined with : → 4 = {1,2} and the weight function of the vertex inclusive local irregularity. The labeling of inclusive local irregularity vertex on the book graph ( 4 ) is as follows. Next it will be investigated that the chromatic numbers of inclusive local irregularities on the graph book

CONCLUSIONS
Based on the results and the discussion, three new theorems were obtained on the topic of inclusive local irregularity vertex coloring in the book graph family. The chromatic number of inclusive local irregularity obtained from the book graph ( 4 ) is 4 and chromatic of from the book graph ( 5 ) is 5. While the chromatic number of inclusive local irregularity in the book graph ( 6 ), there are 2 cases where the chromatic number processed for = 3 is 5 and ≥ 4 is 6.