RAINBOW CONNECTION NUMBER AND TOTAL RAINBOW CONNECTION NUMBER OF AMALGAMATION RESULTS DIAMOND GRAPH( 𝐁𝐫 𝟒 ) AND FAN GRAPH( 𝐅 𝟑 )

. If 𝐺 = 𝑉(𝐺),𝐸(𝐺) be a graph and edge coloring of G is a function 𝑓: 𝐸(𝐺) → 𝐶 , rainbow connection number is the minimum-k coloration of the rainbow on the edge of graph G and denoted by rc(G). Rainbow connection numbers can be applied to the result of operations on some special graphs, such as diamond graphs and fan graphs. Graph operation is a method used to obtain a new graph by combining two graphs. This study performed amalgamation operations to obtain rainbow connection numbers and rainbow-total-connection numbers in diamond graphs ( 𝐵𝑟 4 ) and fan graphs ( 𝐹 3 ) or 𝐴𝑚𝑎𝑙(3𝐵𝑟 4 ∗ 𝐹 3 ,𝑣) . Based on the research, it is obtained that the rainbow-connection number theorem on the amalgamation result of the diamond graph ( 𝐵𝑟 4 ) and fan graph ( 𝐹 3 ) is 𝑟𝑐(𝐺) = 𝑑𝑖𝑎𝑚 with 𝑡 = 4 . Furthermore, the theorem related to the total rainbow-connection number on the amalgamation result of the diamond graph( 𝐵𝑟 4 ) and the fan graph ( 𝐹 3 ), is obtained, namely 𝑡𝑟𝑐(𝐺) = 2𝑑𝑖𝑎𝑚 = 8 with 𝑡 = 4 .


INTRODUCTION
One branch of mathematics that is specifically a study in discrete mathematics and is used to represent discrete objects and the relationships between objects is graph theory.The representation usually has a dot to represent the object, while the object's relationship is expressed with an edge.Previously, graphs were first used to solve the Konigsberg bridge problem in 1736 [1].One of the theories developed in graph theory is Chartrand et al. 2005, which first introduced rainbow connection number, the concept of rainbow connection number.A rainbow connection number is coloring on a graph with the condition that there is a rainbow path on the graph side and two adjacent sides can have the same color.However, the side included in the rainbow path must have a different color, whereas the rainbow path is a path where each side does not have the same color.The rainbow connection number is the k-minimum coloring of the rainbow in graph G.It is denoted by rc(G) [2].
Rainbow connection numbers can be applied to the operating results of some special graphs, such as diamond graphs and fan graphs [3], [4].Graph operation is a method used to obtain a new graph by combining two graphs.There are several forms of operation, including joint, corona, comb, shackle, and amalgamation [5].Several studies have been carried out, namely examining rainbow-connection numbers on the results of amalgamation operations of several special graphs, such as what was done by Fitriani and Salman in 2016 [6].Then, in 2019, several studies were carried out on the amalgamation of diamond graphs, including those carried out by Afifah et al. [7] and Cindy et al. [8].Therefore, from several studies that have been done previously, the author tried to develop rainbow connection numbers by adding amalgamation operations from two special graphs.
In this study, amalgamation operations of two special graphs were performed, including the diamond graph ( 4 ) and the fan graph ( 3 ).The amalgamation operations were performed by agglutinating all terminal points  0 of several graphs   into one point [9].Next, a rainbow coloring will be determined from the amalgamation of a diamond graph( 4 ) and fan graph ( 3 ).This study aimed to determine the rainbow-connection number and the rainbow-totalconnection number from the amalgamation of the diamond graph ( 4 ) and the fan graph ( 3 ), so that two theorems related to rainbow connection numbers, and rainbow-total-connection numbers will be obtained.This study hopefully can contribute to the development of science in the field of graph theory related to rainbow connection numbers and rainbow total connection numbers.

RESEARCH METHOD
In this study, the method used was a literature study research method.The steps used are as follows: 1. Formulate the research problem to be discussed.2. Study and understand literature sources related to rainbow connection number coloring.3. Analyze the problems that had been obtained.Then, patterns and theorems to be proven would be obtained.The steps were as follows: (a) Draw the graph to be studied, namely the diamond and fan graphs.(b) Apply amalgamation operations to diamond and fan graphs.(c) Determine the pattern of rainbow connection numbers and rainbow-total-connection numbers result of amalgamation of diamond and fan graph operations.(d) Proving the rainbow-connection number theorem and the rainbow-total-connection number resulting from the amalgamation of diamond graphs and fan graphs.4. Formulate the conclusions that had been obtained.Furthermore, several definitions and theorems that were used in this study were presented.

Definition 1
If n and t be positive integers with size of  = 4 is an amalgamation of diamond graph and fan graph.A diamond graph with n = 4 denoted by ( 4 ) is a graph with [3]: For more details, the amalgamation of diamond and fan graphs can be seen in Figure 1 below.It can be seen in Figure 1 which is the result of the amalgamation operation of a diamond graph with n = 4 and a fan graph with n=3 for t=4.The amalgamation operation above is carried out by agglutinating all  0 terminal points on three diamond graphs and one fan graph, so that they become one point.. Theorem1 [2] If G is a nontrivially connected graph of size m and () = max {(, )|,  ∈ ()}, so: () ≤ () ≤ () ≤

Rainbow connection numbers Result of Amalgamation of Diamond Graph(〖Br〗_4) and Fan Graph(F_3)
Figure 1 is an amalgamation of three diamond graphs and one fan graph which will then be searched for rainbow connection numbers from each of these graphs so that the following theorem is obtained.:It can be seen in Figure 2, which is the result of coloring the rainbow connection numbers on Amal(3Br 4 * 3 , ), From the coloring, it can be concluded that the result of the amalgamation operation of the diamond graph and the fan graph is that there is a rainbow-connection number coloring.A rainbow coloring is determined by calculating the graph's diameter, then labeling the graph is done on Amal(3Br4* 3 , ).Furthermore, it will be shown in table 1 that for each x and y with (, ) ≥ 2, there is a path x to y, which is a rainbow path.If given a total of 7 colors of the rainbow on G, there will be the same color.Assign a color to the point v, so there will be 6 colors left to color the point and edges on  , ,  , , ,  , ,  , with any point and side coloring.In such a way that the points and edges on the path acquire the same color.Therefore, there is no total rainbow path  , −   .This is against the supposition () ≤ 2() − 1 = 7.Then, it must be () = 2() = 8 for  = 4.It is clear that two points are neighbors ,  ∈ ().There is a rainbow path-total x to ithat is xy.Then, it will be shown in table 2 that for every x and y, there is a rainbow total path.