Rainbow Connection Number on Amalgamation of General Prism Graph

Let G be a nontrivial connected graph, the rainbow-k-coloring of graph G is the mapping of c: E(G) → {1,2,3, ⋯ , k} such that any two vertices from the graph can be connected by a rainbow path (the path with all edges of different colors). The least natural number k of k-edge coloring graph G is called the rainbow connection number of G, denoted by rc(G). For t ∈ N and t ≥ 2, let {P(m,2)i|i ∈ (1,2, ⋯ , t), m ≥ 3} is a collection of prism graph with a fixed vertex v called terminal. The amalgamation of such a prism graph is denoted by Amalt(Pm,2 ) with m ≥ 3. We figured that rc(Amalt(Pm,2)) forms a particular pattern and this research aims to find the formula of such pattern. The result of this research is (Amalt(Pm,2)) = 2 ⌈ m+1 2 ⌉ , t ≥ 2 and m ≥ 3.


INTRODUCTION
The basic idea of graph were first introduced by Leonhard Euler when he attempted to find the solution for the Königsberg bridge problem [1]. The one branch of graph is graph labeling that introduced by Rosa [2] and graph coloring is a special case of this graph. Graph coloring has been one of the more popular branches in mathematics. The rainbow connection number is one of the colorings of graph theory that is interesting to talk about, rainbow connection number is a unique type of coloring because it requires different colors to color the side of the graph. Rainbow connection number can also be considered as something that has been discussed quite often, proven by the number of journals that have talked about this topic. Chartrand et al. [3] first introduced a rainbow connection number.
Let G be a nontrivial connected graph where we define a coloring : ( ) → {1,2, … , }, ∈ ℕ where adjacent edges in G may have the same color. Such coloring is called rainbow coloring in graph G if graph G is rainbow-connected, that is, when any two vertices u, v in G contains a uv rainbow patha path that consists no two edges with the same color. If k color is used, then the coloring c of graph G is said to be a rainbow k-coloring with k being its rainbow connection number in graph G, denoted as ( ) [4].
Charttrand et al. [3] determined rainbow connection number in some special types of graphs such as the wheel graph, the k-partite complete graph, the tree graph, and the cycle graph. Some research in rainbow connection number is Kumala and Salman [4] discuss about the rainbow connection number for the flower graph ( , ), Darmawan [5] discussed about the rainbow connection number on some special graphs, and Palupi et al. [6] determined the rainbow connection number in prism graph 3,2 amalgamation which resulted ( ) = 4. Similar to Palupi et al. [6], we will explore rainbow connection number in prism graph ,2 amalgamation to obtain a more general theorem.

PRISM GRAPH
Prism Graph is the Cartesian product × , that is, a cycle graph with vertices and a path , 1 ≤ ≤ − 1} act as its vertices and edges set respectively [6]. Fitriani and Salman [7] show that the lower bound and upper bound of a connection for any graph. They proved that the lower limit of the number of rainbow connection in graphs is the diameter of the graph itself. Figure 1 illustrates the Prism Graph 3,2 .

RESULT AND DISCUSSION
We begin to pay attention to the diameter of a circle graph . Figure 2 illustrates that the diameter of graph C with ≥ 3 will never exceed half of the circumference in its cycle and ⌈ m−1 2 ⌉ to be exact.
Proof. Graph ,2 is the product of × 2 , therefore, its figure will form a double-layered Graph with two layers connected. See Figure 3. From this figure, the diameter of the prism graph is a distance from one point of the outer graph to the inner graph . Hence it needs an additional one side from ( ). Therefore we can write:  Proof. As we know, ( ,2 ) is a group of a minimum of 2 prism graphs connected by a terminal vertex. Because the longest distance of any two points in a said graph has to go pass its terminal vertex as illustrated in Figure 4, then the diameter is twice of the diameter of a prism graph, hence the diameter of this graph is 2 × ( ,2 ). □

Rainbow Connection Number on Prism Graph ( , ) Amalgamation
See Picture 5. From this figure, we achieve that ( ( (3,2) )) = 4. Note that the value of t will not affect the value of the rainbow connection number because every path will pass the terminal point.

Rainbow Connection Number( ( , ))
The next theorem is the result of the coloring of rainbow connection number in prism graph , amalgamations. Proof. It will be shown that there is a rainbow connection number with a value of 2 ⌈ It will be shown that P(m) is true for every m ℕ with m ≥ 3 via the principle of mathematical induction.
(ii) Assume that ( ) is true. (iii) It will be shown that ( + 1) is true.
For example, Figure 6 is a pair of the farthest side from . From this illustration, it is clear that any additional color is not needed when we add an extra layer. Moreover, if we look at it from the newly formed points in ( 2 , 2 ) tracks to the already existing points can be found without repeating a color, thus ( ( 2 +1,2 )) = 2 ⌈ 2 +1+1 2 ⌉ = 2 + 2.  For example, Figure 7 is a pair of the farthest side from . From this illustration, clearly that 2 additional colors will be needed and if we look at it from the newly formed points in ( 2 , 2 ) a track to the already existing points can be found without repeating a color, thus it has to be ( ( 2 ,2 )) = 2 ⌈ 2 +1 2 ⌉ = 2 + 2.

CONCLUSION
We get a general formula using the theorems that we got to obtain rainbow connection number on prism graph ,2 is ( ( ,2 )) = 2 ⌈