RAINBOW VERTEX-CONNECTION NUMBER ON COMB PRODUCT OPERATION OF CYCLE GRAPH (𝐂 𝟒 ) AND COMPLETE BIPARTITE GRAPH (𝐊 𝟑,𝐧 )

ABSTRACT


INTRODUCTION
One of the subjects in the field of combinatorial mathematics is graph theory, which has been widely used to solve a problem. The use of graphs is to present discrete objects and the relationships between them [1]. Graph theory gets a lot of attention because the models used can be applied in everyday life. Among them such as scheduling transportation departures, scheduling courses, determining the shortest trajectory, and many more. Some theories support this, graphs theory solving and analysis of automated electric networks [2], and applications of graphs theory for solving electrical circuits [3].
Graph labeling is one of the topics in graph theory. Graph labeling is presented by vertex and edge along with a set of natural number members called labels. There are three types of labeling on a graph ( ), namely point labeling, side labeling, and total labeling [4].
Graph coloring is a special case in graph labeling. Giving color to the vertex or edge of a graph is the definition of graph coloring. Three types of coloring are obtained in graphs, namely vertex coloring, edge coloring, and field coloring. Vertex coloring gives color to a vertex, where each of the two adjacent vertex acquires a different color. Edge coloring gives color to the edge, where each of the two adjacent edges obtains a different color. Field coloring gives color to the field, where each of the two adjacent fields obtains a different color [5] [6]. The rainbow connection is a development of graph coloring. A rainbow connection is the coloring of the edge of the graph, where each vertex on the graph is connected by a trajectory that has a different edge color. Rainbow connection number of a graph ( ), symbolized by ( ) is a number that states the number of minimum coloring in a graph ( ) [7] [8].
Krivelevich and Yuster began introducing the rainbow vertex connection in 2009, that was a development of the rainbow connection concept [9]. Rainbow vertex connection is the coloring of the vertex of a graph, so that each vertex on the graph is connected by a trajectory that has an interior vertex with different colors. Rainbow vertex connection number symbolized by ( ) is a number that expresses the number of minimum coloring in a graph ( ). Numbers connected to rainbow vertex can also be applied to graphs that are the result of operations. One method for obtaining a new graph form is to perform an operation between two graphs. There are various types of operations in the graph, such as join (⨁), combined (∪), cartesian (×), corona (⨀), and comb (⊳) operations.
In recent studies, we have studied and developed rainbow-connected numbers as a result of surgery. The rainbow connection numbers of the sum and multiplication operations of graphs [10] [11]. The determination of rainbow connection numbers in the results of cartesian product operations against circle graphs and bipartite graphs complete with track graphs [12]. The total rainbow connected numbers resulting from comb operations on graph cycles and graph paths [13]. Rainbow connection number of comb product of graphs. This research give sharp lower and upper bounds for the rainbow connection number of comb product between two connected graphs [14]. The numbers connected to strong rainbow vertex on the graph resulting from edge comb operations [15].
Based on the background description above, this study discusses the number of connected rainbow vertex by using vertex comb operation, where the graph to be operated is the cycle graph ( 4 ) and the complete bipartite graph ( 3, ).

RESEARCH METHODS
This research uses literature study research methods (library research). In this study, a study of books, textbooks, journals, and scientific articles on the number of rainbow vertex connected to the results of comb ( ) operations with the aim of obtaining information and methods used in the completion of this study.

RESULTS AND DISCUSSION
In this section, we determine rainbow-vertex connection number resulting from comb product operation of cycle graph and complete bipartite graph ( ⊳ , ) & ( , ⊳ ).
Definition 1: Suppose ≥ 2. The 4 graph is a cycle graph with 4 vertices. The 3, graph is a complete bipartite graph with a size of 3, . Thus, the comb operation for the Cycle graph and the Complete Bipartite graph is denoted with ( 4 ⊳ 3, ). Suppose ( 4 ⊳ 3, ) is the graph , then graph is formed by the set of vertex and edge defined as follows.
Graph image of comb operation 4 ⊳ 3,2 is shown in Figure 1.

Rainbow Vertex-Connection Number on Comb Product Operation of Cycle Graph and Complete Bipartite Graph ( 4 ⊳ 3, )
In this section, we describe Rainbow Vertex-Connection Number on Comb Product Operation of Cycle Graph and Complete Bipartite.
Based on the trajectory of the vertex rainbow in Table 1, the theorem ( 4 ⊳ 3, ) = 5 is proven.

Comb Complete Bipartite Graph and Cycle Graph ( 3, ⊳ 4 )
In this section, we describe comb complete bipartite graph and cycle graph and constructing figure Rainbow Vertex-Connection Number of Graph 3,2 ⊳ 4 .

Definition 2:
Suppose ≥ 2. The 3, graph is a complete bipartite graph with a size of 3, . The 4 graph is a cycle graph with 4 vertices. Thus, the comb operation for the Complete Bipartite graph and the Cycle graph is denoted with ( 3, ⊳ 4 ). Suppose ( 3, ⊳ 4 ) is the graph , then graph is formed by the set of vertex and edge defined as follows.
Graph image of comb operation 3,2 ⊳ 4 is shown in Figure 3.  It will then be shown that for each and in ( 3, ⊳ 4 ) there is an − rainbow vertex-connection.   Based on the trajectory of the vertex rainbow in Table 2, the theorem ( 3,2 ⊳ 4 ) = 3 − 1 proven.
Noticed the vertex 6 cannot be colored 1. If 6 is given color 1, then there will be a trajectory that is not rainbow, namely the 1,1 , 1 , 6 , 6,1 . Furthermore, the 6 vertex cannot be colored 2. If given color 2, then there will be a track that is not rainbow, namely the track 2,1 , 2 , 6 , 6,1 . Furthermore, the 6 vertex cannot be colored 3. If given color 3, then there will be a trajectory that is not rainbow, namely the track 3,1 , 3 , 6 , 6,1 . Furthermore, the 6 vertex cannot be colored 4. If given color 4, then there will be a track that is not rainbow, namely the 4,1 , 4 , 2 , 6 , 6,1 . Furthermore, the 6 vertex cannot be colored + 2 = 5. If given the color + 2 = 5, then there will be a path that is not rainbow, namely the 5,1 , 5 , 2 , 6 , 6,1 .