On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion

For a bijective function g: E(G) → {1, 2,3, ⋯ , |E(G)|}, the associated weight of a vertex v ∈ V(G) under g is wg(v) = Σe∈E(v)g(e), where E(v) is the set of vertices incident to v. The function g is called a vertex-antimagic edge labeling if every vertex has distinct weight. A path P in the edgelabeled graph G is said to be a rainbow path if for any two vertices x and x′, all internal vertices in the path x − x′ have different weight. If for every two vertices x and y of G, there exists a rainbow x − y path, then g is called a rainbow vertex antimagic labeling of G. When we assign each edge xy with the color of the vertex weight wg(v), thus we say the graph G admits a rainbow vertex antimagic coloring. The smallest number of colors taken over all rainbow colorings induced by rainbow vertex antimagic labelings of G is called rainbow vertex antimagic connection number of G, denoted by rvac(G). In this paper, we initiate to determine the rainbow vertex antimagic connection number of graphs, namely path (Pn), wheel (Wn), friendship (Fn), and fan (Fn).


INTRODUCTION
We consider a graph ( , ) in this paper are simple, connected and un-directed graph, where and are respectively a vertex set and edge set of [1]. The Rainbow coloring problem has been studied by many researchers since many years ago. Many good results has been published in some reputable journal [2]. Thus, it has given many contributions in graph theory research of interest. There are many types of rainbow coloring, namely rainbow (edge) coloring, rainbow vertex coloring, strong rainbow edge/vertex coloring. The minimum number of colors for which an edge (vertex) coloring exists such that the graph is rainbow connected is called the rainbow connection number, denoted by ( ) for edge coloring and the rainbow vertex connection number, denoted by ( ) for vertex coloring, see [3]- [10] for detail. Krivelevich and Yuster [6] gave the lower bound for Meanwhile, In 2003, Hartsfield and Ringel [11] defined antimagic graphs. A graph is called antimagic if there exists a bijection : ( ) → {1,2, ⋯ , } such that the weights of all vertices are distinct [12] . The vertex weight of a vertex under , ( ), is the sum of labels of edges incident with , that is, ( ) = ∑ ( ) ∈ ( ) . In this case, is called an antimagic labeling. There many results were found for antimagicness of graph. There are extension types of vertex antimagic labeling, namely total vertex antimagic labeling, super total vertex antimagic labeling, ( , )-vertex antimagic labeling, super ( , )-vertex antimagic labeling. For detail, see Galian Dynamic Survey of Graph Labeling [13] .
In this study, we initiate to combine the two notion, namely rainbow coloring and antimagic labeling [14] [15]. We name for this combination as rainbow vertex antimagic coloring. For a bijective function : ( ) → {1, 2,3, ⋯ , | ( )|}, the associated weight of a vertex ∈ ( ) under is ( ) = Σ ∈ ( ) ( ), where ( ) is the set of vertices incident to . The function is called a vertex-antimagic edge labeling if every vertex has distinct weight. A path in the edge-labeled graph is said to be a rainbow path if for any two vertices and ′, all internal vertices in the path − ′ have different weight. If for every two vertices and of , there exists a rainbow − path, then is called a rainbow vertex antimagic labeling of . When we assign each edge with the color of the vertex weight ( ), thus we say the graph admits a rainbow vertex antimagic coloring. The rainbow vertex antimagic connection number of , denoted by ( ), is the smallest number of colors taken over all rainbow colorings induced by rainbow vertex antimagic labelings of .
To determine the rainbow vertex antimagic connection number of any graph is considered to be hard problem. Even, this study fall into NP-hard problem. In this paper, we initiate to determine the rainbow vertex antimagic connection number of graphs, namely path ( ), wheel ( ), friendship (ℱ ), and fan ( ) as well as fix the lower bound ( ) of any graph.

METHODS
This research includes deductive analytic methods. The procedures to obtain the rainbow vertex antimagic connection number of are as follows. 1. Define a graph . 2. Determine the cardinality of graph by obtaining the order and size of graph . 3. Determine the lower bound of ( ) by using the obtained remark of sharpest lower bound. 4. Determine the upper bound of ( ) by constructing the bijective function, compute the vertex weight using ( ) = Σ ∈ ( ) ( ), and show that every two different vertices of satisfy the rainbow vertex antimagic coloring. 5. If the upper bound attains the lower bound, then we obtain the ( ). If the upper bound does not attain the lower bound, then we return to determine the upper bound of ( ). 6. Finally we can construct a new theorem and its proof after we obtain the rainbow vertex antimagic connection number of graph .

On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion
Marsidi 66

RESULTS AND DISCUSSION
In this section we have several theorems on the rainbow vertex antimagic coloring. We determine the minimum color taken to the graph such that it has rainbow vertex antimagic coloring. Since we determine the minimum colors such that has rainbow vertex antimagic coloring, then the lower bound of rainbow vertex antimagic connection number of graph is at least and equal to rainbow vertex connection number. The lower bound of rainbow vertex antimagic connection number of any graph is mathematically written in the Remark 1.

Theorem 1
If be a path graph of order and ≥ 3, then Proof. Let be a path graph with vertex set ( ) = { 1 , 2 , 3 , ⋯ , } and edge set We divide into two cases to prove the rainbow vertex antimagic connection number as follows. Furthermore, we show that every two different vertices of is rainbow vertex antimagic coloring. Suppose that ∈ ( ), refer to the vertex weight the rainbow vertex path is shown in Table 1.  Since the vertex has degree of much greater than the others, it must have a different vertex weight than the others. The vertex weight of is the sum of labels of edges which incident to . From this condition, such that we have ( ) ≥ 2. We divide into two cases to show the upper bound of the rainbow vertex antimagic connection number of as follows.

On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion
Marsidi 68
if ≡ 0(mod 2) ( ) = 2 + 1 − From the edge labels above, we have the vertex weights in the following: From the vertex weights above, it is easy to see that the different weight is 2.
From the edge labels above, we have the vertex weights in the following.
Furthermore, we show that every two different vertices of is rainbow vertex antimagic coloring. Suppose that , ∈ ( ), refer to the vertex weight the rainbow vertex − path is shown in Table 2. Hence, the vertex coloring of is rainbow vertex antimagic coloring. Thus, we obtain ( ) = 2 if ≡ 1(mod 2) and 2 ≤ ( ) ≤ 3 if ≡ 0(mod 2). ∎

Theorem 3
If ℱ be a friendship graph of order 2 + 1 and ≥ 3, then (ℱ ) = 3.  others, it must have a different vertex weight than the others. The vertex weight of is the sum of labels of edges which incident to . In the other hand, the vertex and are adjacent, such that based on the edge labeling it can not receive the same weight. From this condition, such that we have (ℱ ) ≥ 3. Furthermore, to show the upper bound we construct the bijective function of edge labels.

Proof. Let
From the edge labels above, we have the vertex weights in the following.
( ) = 2 + 1 ( ) = 4 + 1 ( ) = 3 2 + From the vertex weight above, it is easy to see that the different weight is 3. Furthermore, we show that every two different vertices of ℱ is rainbow vertex antimagic coloring. Suppose that , ∈ (ℱ ), refer to the vertex weight the rainbow vertex − path is shown in Table 3. Hence, the vertex coloring of ℱ is rainbow vertex antimagic coloring. Thus, we obtain (ℱ ) is 3 . ∎

Theorem 4
If be a fan graph +1 and ≥ 3, then ( ) = 2 if ≡ 1(mod 2) and 2 ≤ ( ) ≤ 3 if ≡ 0(mod 2). Since the vertex has degree of much greater than the others, it must have a different vertex weight than the others. The vertex weight of is the sum of labels of edges which incident to . From this condition, such that we have ( ) ≥ 2. We divide into two cases to show the upper bound of the antimagic rainbow connection number of as follows.
From the edge labels above, we have the vertex weights in the following.
if 1 ≤ ≤ − 1 From the edge labels above, we have the vertex weights in the following.
From the vertex weight above, it is easy to see that the different weight is 3. Furthermore, we show that every two different vertices of is rainbow vertex antimagic coloring. Suppose that , ∈ ( ), refer to the vertex weight the rainbow vertex − path is shown in Table 4. Hence, the vertex coloring of is rainbow vertex antimagic coloring. Thus, we obtain ( ) = 2 if ≡ 1(mod 2) and 2 ≤ ( ) ≤ 3 if ≡ 0(mod 2). ∎ The illustration of antimagic rainbow edge labeling can be seen in Figure 1. Based on the Figure 1, we know that wheel graph 17 satisfy the rainbow vertex antimagic coloring and rainbow vertex antimagic connection number of 17 is 2.

On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion
Marsidi 71

CONCLUSIONS
We have obtained the exact values of rainbow vertex antimagic connection number of some connected graphs, namely path ( ), wheel ( ), friendship (ℱ ), and fan ( ). However, since obtaining rainbow vertex antimagic connection number of graph is considered to be NP-complete problem, the characterization of the exact value of ( ) for any family graph is still widely open. Therefore, we propose the following open problems as follows. 1. Determine the exact value of rainbow vertex antimagic connection number of graphs apart from those families. 2. Determine the exact value of rainbow vertex antimagic connection number of any operation graphs.