On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion CAUCHY – Jurnal Matematika Murni dan Aplikasi Volume 7(1) (2021), Pages 64-72 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: Juni 30, 2021 Reviewed: August 12, 2021 Accepted: October 20. 2021 DOI: https://doi.org/10.18860/ca.v7i1.12796 On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion Marsidi1, Ika Hesti Agustin2, Dafik3, Elsa Yuli Kurniawati4 1Department of Mathematics Education, Universitas PGRI Argopuro Jember, Indonesia 2Department of Mathematics, University of Jember, Indonesia 3Department of Mathematics Education, University of Jember, Indonesia 4CGANT, University of Jember, Indonesia Email: marsidiarin@gmail.com, ikahesti.fmipa@unej.ac.id, d.dafik@unej.ac.id, elsayuli@unej.ac.id ABSTRACT For a bijective function 𝑔: 𝐸(𝐺) β†’ {1, 2,3, β‹― , |𝐸(𝐺)|}, the associated weight of a vertex 𝑣 ∈ 𝑉(𝐺) under 𝑔 is 𝑀𝑔(𝑣) = Ξ£π‘’βˆˆπΈ(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 edge- labeled graph 𝐺 is said to be a rainbow path if for any two vertices π‘₯ and π‘₯β€², all internal vertices in the path π‘₯ βˆ’ π‘₯β€² have different weight. If for every two vertices π‘₯ and 𝑦 of 𝐺, there exists a rainbow π‘₯ βˆ’ 𝑦 path, then 𝑔 is called a rainbow vertex antimagic labeling of 𝐺. When we assign each edge π‘₯𝑦 with the color of the vertex weight 𝑀𝑔(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 π‘Ÿπ‘£π‘Žπ‘(𝐺). In this paper, we initiate to determine the rainbow vertex antimagic connection number of graphs, namely path (𝑃𝑛), wheel (π‘Šπ‘› ), friendship (ℱ𝑛), and fan (𝐹𝑛). Keywords: antimagic labeling; rainbow vertex coloring; rainbow vertex antimagic coloring; rainbow vertex antimagic connection number. INTRODUCTION We consider a graph 𝐺(𝑉, 𝐸) in this paper are simple, connected and un-directed graph, where 𝑉 and 𝐸 are respectively a vertex set and edge set of 𝐺 [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 𝐺 is rainbow connected is called the rainbow connection number, denoted by π‘Ÿπ‘(𝐺) for edge coloring and the rainbow vertex connection number, denoted by π‘Ÿπ‘£π‘(𝐺) for vertex coloring, see [3]–[10] for detail. Krivelevich and Yuster [6] gave the lower bound for π‘Ÿπ‘£π‘(𝐺), namely π‘Ÿπ‘£π‘(𝐺) β‰₯ π‘‘π‘–π‘Žπ‘š(𝐺) – 1, where π‘‘π‘–π‘Žπ‘š(𝐺) is the diameter of graph 𝐺. An easy observation is that if 𝐺 has an order n, then π‘Ÿπ‘£π‘(𝐺) ≀ 𝑛 βˆ’ 2 and π‘Ÿπ‘£π‘(𝐺) = 0 if and only if 𝐺 is a complete graph. Notice that π‘Ÿπ‘£π‘(𝐺) β‰₯ π‘‘π‘–π‘Žπ‘š(𝐺) βˆ’ 1 with equality if the diameter of 𝐺 is 1 or 2. https://doi.org/10.18860/ca.v7i1.12796 mailto:marsidiarin@gmail.com mailto:ikahesti.fmipa@unej.ac.id mailto:d.dafik@unej.ac.id mailto:elsayuli@unej.ac.id On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion Marsidi 65 Meanwhile, In 2003, Hartsfield and Ringel [11] defined antimagic graphs. A graph 𝐺 is called antimagic if there exists a bijection 𝑓: 𝐸(𝐺) β†’ {1,2, β‹― , π‘ž} such that the weights of all vertices are distinct [12] . The vertex weight of a vertex 𝑣 under 𝑓, 𝑀𝑓 (𝑣), is the sum of labels of edges incident with 𝑣, that is, 𝑀𝑓 (𝑣) = βˆ‘ 𝑓(𝑒𝑣)π‘’π‘£βˆˆπΈ(𝐺) . In this case, 𝑓 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, (π‘Ž, 𝑑)-vertex antimagic labeling, super (π‘Ž, 𝑑)-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 𝑔: 𝐸(𝐺) β†’ {1, 2,3, β‹― , |𝐸(𝐺)|}, the associated weight of a vertex 𝑣 ∈ 𝑉(𝐺) under 𝑔 is 𝑀𝑔(𝑣) = Ξ£π‘’βˆˆπΈ(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 edge-labeled graph 𝐺 is said to be a rainbow path if for any two vertices π‘₯ and π‘₯β€², all internal vertices in the path π‘₯ βˆ’ π‘₯β€² have different weight. If for every two vertices π‘₯ and 𝑦 of 𝐺, there exists a rainbow π‘₯ βˆ’ 𝑦 path, then 𝑔 is called a rainbow vertex antimagic labeling of 𝐺. When we assign each edge π‘₯𝑦 with the color of the vertex weight 𝑀𝑔(𝑣), thus we say the graph 𝐺 admits a rainbow vertex antimagic coloring. The rainbow vertex antimagic connection number of 𝐺, denoted by π‘Ÿπ‘£π‘Žπ‘(𝐺), is the smallest number of colors taken over all rainbow colorings induced by rainbow vertex antimagic labelings of 𝐺. 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 (𝑃𝑛 ), wheel (π‘Šπ‘›), friendship (ℱ𝑛), and fan (𝐹𝑛 ) as well as fix the lower bound π‘Ÿπ‘£π‘Žπ‘(𝐺) 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 𝐺. 2. Determine the cardinality of graph 𝐺 by obtaining the order and size of graph 𝐺. 3. Determine the lower bound of π‘Ÿπ‘£π‘Žπ‘(𝐺) by using the obtained remark of sharpest lower bound. 4. Determine the upper bound of π‘Ÿπ‘£π‘Žπ‘(𝐺) by constructing the bijective function, compute the vertex weight using 𝑀𝑔(𝑣) = Ξ£π‘’βˆˆπΈ(𝑣)𝑔(𝑒), and show that every two different vertices of 𝐺 satisfy the rainbow vertex antimagic coloring. 5. If the upper bound attains the lower bound, then we obtain the π‘Ÿπ‘£π‘Žπ‘(𝐺). If the upper bound does not attain the lower bound, then we return to determine the upper bound of π‘Ÿπ‘£π‘Žπ‘(𝐺). 6. Finally we can construct a new theorem and its proof after we obtain the rainbow vertex antimagic connection number of graph 𝐺. 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 𝐺 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. Remark 1 Let 𝐺 be a connected graph, π‘Ÿπ‘£π‘Žπ‘(𝐺) β‰₯ π‘Ÿπ‘£π‘(𝐺). Theorem 1 If 𝑃𝑛 be a path graph of order 𝑛 and 𝑛 β‰₯ 3, then π‘Ÿπ‘£π‘Žπ‘(𝑃𝑛) = { 3, 𝑛 = 3,4 𝑛 βˆ’ 2, 𝑛 β‰₯ 5 Proof. Let 𝑃𝑛 be a path graph with vertex set 𝑉(𝑃𝑛) = {𝑣1, 𝑣2, 𝑣3, β‹― , 𝑣𝑛 } and edge set 𝐸(𝑃𝑛 ) = {𝑣𝑖 𝑣{𝑖+1}: 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1}. The diameter of 𝑃𝑛 is 𝑛 βˆ’ 1. We divide into two cases to prove the rainbow vertex antimagic connection number as follows. Case 1. For 𝑃𝑛 , 𝑛 = 3,4 Path graph 𝑃𝑛 , 𝑛 = 3 have two edges. If we give labels on it, it gives three different weights on its edges exactly. It concludes that the rainbow vertex antimagic connection number of 𝑃3 is 3. Furthermore for 𝑃4, we determine the all permutation of edge labeling on 𝑃4. Let 𝑒1, 𝑒2, 𝑒3 are the edges of 𝑃4, thus there are six possibilities of edge labeling on 𝑃4 as follows. 1). If 𝑒1 = 1, 𝑒2 = 2, 𝑒3 = 3, then 𝑀𝑑(𝑣1) = 1, 𝑀𝑑(𝑣2) = 3, 𝑀𝑑(𝑣3) = 5, 𝑀𝑑(𝑣4) = 3. 2). If 𝑒1 = 1, 𝑒2 = 3, 𝑒3 = 2, then 𝑀𝑑(𝑣1) = 1, 𝑀𝑑(𝑣2) = 4, 𝑀𝑑(𝑣3) = 5, 𝑀𝑑(𝑣4) = 2. 3). If 𝑒1 = 2, 𝑒2 = 1, 𝑒3 = 3, then 𝑀𝑑(𝑣1) = 2, 𝑀𝑑(𝑣2) = 3, 𝑀𝑑(𝑣3) = 4, 𝑀𝑑(𝑣4) = 3. 4). If 𝑒1 = 2, 𝑒2 = 3, 𝑒3 = 1, then 𝑀𝑑(𝑣1) = 2, 𝑀𝑑(𝑣2) = 5, 𝑀𝑑(𝑣3) = 4, 𝑀𝑑(𝑣4) = 1. 5). If 𝑒1 = 3, 𝑒2 = 1, 𝑒3 = 2, then 𝑀𝑑(𝑣1) = 3, 𝑀𝑑(𝑣2) = 4, 𝑀𝑑(𝑣3) = 3, 𝑀𝑑(𝑣4) = 2. 6). If 𝑒1 = 3, 𝑒2 = 2, 𝑒3 = 1, then 𝑀𝑑(𝑣1) = 3, 𝑀𝑑(𝑣2) = 5, 𝑀𝑑(𝑣3) = 3, 𝑀𝑑(𝑣4) = 1. Based on edge labelings and vertex weights above, it is easy to determine the rainbow vertex antimagic connection number of 𝑃4 at least 3. Thus π‘Žπ‘Ÿπ‘£π‘(𝑃4) = 3. Case 2. For 𝑃𝑛 , 𝑛 β‰₯ 5 Based on Remark 1, we have π‘Ÿπ‘£π‘Žπ‘(𝑃𝑛) β‰₯ π‘Ÿπ‘£π‘(𝑃𝑛 ) = π‘‘π‘–π‘Žπ‘š(𝑃𝑛 ) βˆ’ 1 = 𝑛 βˆ’ 1 βˆ’ 1 = 𝑛 βˆ’ 2. Furthermore, to show the upper bound we construct the bijective function of edge labels. We have two conditions, namely for 𝑛 ≑ 1(mod 2) and 𝑛 ≑ 0(mod 2). For 𝑛 ≑ 1(mod 2), we have 𝑔(𝑣1𝑣2) = 3 𝑔(𝑣2𝑣3) = 1 𝑔(𝑣3𝑣4) = 2 𝑔(π‘£π‘›βˆ’1𝑣𝑛 ) = 4 𝑔(𝑣𝑖 𝑣𝑖+1) = 𝑖 + 1: 4 ≀ 𝑖 ≀ 𝑛 βˆ’ 2 From the edge labels above, we have the vertex weight as follows. For 𝑃5, we have 𝑀(𝑣1, 𝑣2, 𝑣3, 𝑣4, 𝑣5) = (3,4,3,6,4). For 𝑃𝑛 : 𝑛 β‰₯ 6, we have On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion Marsidi 67 𝑀(𝑣1) = 3 𝑀(𝑣2) = 4 𝑀(𝑣3) = 3 𝑀(𝑣4) = 7 𝑀(𝑣𝑖 ) = 2𝑖 + 1: 5 ≀ 𝑖 ≀ 𝑛 βˆ’ 2 𝑀(π‘£π‘›βˆ’1) = 𝑛 + 3 𝑀(𝑣𝑛) = 4 For 𝑛 ≑ 0(mod 2), we have 𝑔(𝑣1𝑣2) = 3 𝑔(𝑣2𝑣3) = 1 𝑔(𝑣3𝑣4) = 2 𝑔(𝑣𝑖 𝑣𝑖+1) = 𝑖: 4 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 From the edge labels above, we have the vertex weights in the following: 𝑀(𝑣1) = 3 𝑀(𝑣2) = 4 𝑀(𝑣3) = 3 𝑀(𝑣4) = 6 𝑀(𝑣𝑖 ) = 2𝑖 βˆ’ 1: 5 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 𝑀(𝑣𝑛) = 𝑛 βˆ’ 1 From the vertex weight above, it is easy to see that the different weight is 𝑛 βˆ’ 2. It concludes that the rainbow vertex antimagic connection number of 𝑃𝑛 : 𝑛 = {3,4} is 3 and the rainbow vertex antimagic connection number of 𝑃𝑛 : 𝑛 β‰₯ 5 is 𝑛 βˆ’ 2. Furthermore, we show that every two different vertices of 𝑃𝑛 is rainbow vertex antimagic coloring. Suppose that 𝑣 ∈ 𝑉(𝑃𝑛 ), refer to the vertex weight the rainbow vertex path is shown in Table 1. Table 1. The Rainbow Vertex Path of 𝑃𝑛 Case 𝒗 𝒗 Rainbow Vertex Coloring 1 𝑣1 𝑣𝑛 𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑖 , … , π‘£π‘›βˆ’1 Hence, the vertex coloring of 𝑃𝑛 is rainbow vertex antimagic coloring. Thus, we obtain π‘Žπ‘Ÿπ‘£π‘(𝑃𝑛) is 3 for 𝑛 = 3,4 and π‘Žπ‘Ÿπ‘£π‘(𝑃𝑛 ) is 𝑛 βˆ’ 2 for 𝑛 β‰₯ 5. ∎ Theorem 2 If π‘Šπ‘› be a wheel graph of order 𝑛 + 1 and 𝑛 β‰₯ 3, then π‘Ÿπ‘£π‘Žπ‘(π‘Šπ‘›) = 2 if 𝑛 ≑ 1(mod 2) and 2 ≀ π‘Ÿπ‘£π‘Žπ‘(π‘Šπ‘›) ≀ 3 if 𝑛 ≑ 0(mod 2). Proof. Let π‘Šπ‘› be a wheel graph with vertex set 𝑉(π‘Šπ‘›) = {𝐴, π‘₯1, π‘₯2, π‘₯3, β‹― , π‘₯𝑛} and edge set 𝐸(π‘Šπ‘›) = {𝐴π‘₯𝑖 : 1 ≀ 𝑖 ≀ 𝑛} βˆͺ {π‘₯𝑖 π‘₯{𝑖+1}: 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1} βˆͺ {π‘₯π‘›βˆ’1π‘₯1}. The diameter of π‘Šπ‘› is 2. Based on Remark 1, we have π‘Ÿπ‘£π‘Žπ‘(π‘Šπ‘›) β‰₯ π‘Ÿπ‘£π‘(π‘Šπ‘›) = π‘‘π‘–π‘Žπ‘š(π‘Šπ‘›) βˆ’ 1 = 2 βˆ’ 1 = 1. Since the vertex 𝐴 has degree of much greater than the others, it must have a different vertex weight than the others. The vertex weight of 𝐴 is the sum of labels of edges which incident to 𝐴. From this condition, such that we have π‘Ÿπ‘£π‘Žπ‘(π‘Šπ‘›) β‰₯ 2. We divide into two cases to show the upper bound of the rainbow vertex antimagic connection number of π‘Šπ‘› as follows. On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion Marsidi 68 Case 1. For π‘Šπ‘›, 𝑛 ≑ 1(mod 2) To show the upper bound of (π‘Šπ‘›): 𝑛 ≑ 1(mod 2) , we construct the bijective function of edge labels. 𝑔(π‘₯𝑖 π‘₯𝑖+1) = { 𝑖 + 1 2 , if 𝑖 ≑ 1(mod 2) ⌈ 𝑛 2 βŒ‰ + 𝑖 2 , if 𝑖 ≑ 0(mod 2) 𝑔(𝐴π‘₯𝑖 ) = 2𝑛 + 1 βˆ’ 𝑖 From the edge labels above, we have the vertex weights in the following: 𝑀(π‘₯𝑖 ) = 2𝑛 + 1 + ⌈ 𝑛 2 βŒ‰ 𝑀(𝐴) = 𝑛 2 (3𝑛 + 1) From the vertex weights above, it is easy to see that the different weight is 2. Case 2. For π‘Šπ‘›, 𝑛 ≑ 0(mod 2) To show the upper bound of π‘Ÿπ‘£π‘Žπ‘(π‘Šπ‘›): 𝑛 ≑ 0(mod 2), we construct the bijective function of edge labels. 𝑔(π‘₯𝑖 π‘₯𝑖+1) = { 𝑖 + 1 2 , if 𝑖 ≑ 1(mod 2) ⌈ 𝑛 2 βŒ‰ + 𝑖 2 , if 𝑖 ≑ 0(mod 2) 𝑔(𝐴π‘₯𝑖 ) = 2𝑛 + 1 βˆ’ 𝑖 From the edge labels above, we have the vertex weights in the following. 𝑀(π‘₯1) = 3𝑛 + 1 𝑀(π‘₯𝑖 ) = 2𝑛 + 1 + ⌈ 𝑛 2 βŒ‰ 𝑀(𝐴) = 𝑛 2 (3𝑛 + 1) 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 π‘Šπ‘› is rainbow vertex antimagic coloring. Suppose that π‘₯, 𝑦 ∈ 𝑉(π‘Šπ‘›), refer to the vertex weight the rainbow vertex π‘₯ βˆ’ 𝑦 path is shown in Table 2. Table 2. The Rainbow Vertex of π‘₯ βˆ’ 𝑦 Path of π‘Šπ‘› Case 𝒙 π’š Rainbow Vertex Coloring 𝒙 βˆ’ π’š 1 π‘₯𝑖 𝐴 π‘₯𝑖 , 𝐴 2 π‘₯𝑖 π‘₯𝑖 π‘₯𝑖 , 𝐴, π‘₯𝑖 Hence, the vertex coloring of π‘Šπ‘› is rainbow vertex antimagic coloring. Thus, we obtain π‘Ÿπ‘£π‘Žπ‘(π‘Šπ‘›) = 2 if 𝑛 ≑ 1(mod 2) and 2 ≀ π‘Ÿπ‘£π‘Žπ‘(π‘Šπ‘›) ≀ 3 if 𝑛 ≑ 0(mod 2). ∎ Theorem 3 If ℱ𝑛 be a friendship graph of order 2𝑛 + 1 and 𝑛 β‰₯ 3, then π‘Ÿπ‘£π‘Žπ‘(ℱ𝑛) = 3. Proof. Let ℱ𝑛 be a friendship graph with vertex set 𝑉(ℱ𝑛) = {𝐴} βˆͺ {π‘₯1, π‘₯2, π‘₯3, … , π‘₯𝑛 } βˆͺ {𝑦1, 𝑦2, 𝑦3, … , 𝑦𝑛 } and edge set 𝐸(ℱ𝑛) = {𝐴π‘₯𝑖 ; 1 ≀ 𝑖 ≀ 𝑛} βˆͺ {𝐴𝑦𝑖 ; 1 ≀ 𝑖 ≀ 𝑛} βˆͺ {π‘₯𝑖 𝑦𝑖 ; 1 ≀ 𝑖 ≀ 𝑛}. The diameter of ℱ𝑛 is 2. Based on Remark 1, we have π‘Ÿπ‘£π‘Žπ‘(ℱ𝑛) β‰₯ π‘Ÿπ‘£π‘(ℱ𝑛) = π‘‘π‘–π‘Žπ‘š(ℱ𝑛) βˆ’ 1 = 2 βˆ’ 1 = 1. Since the vertex 𝐴 has degree of much greater than the On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion Marsidi 69 others, it must have a different vertex weight than the others. The vertex weight of 𝐴 is the sum of labels of edges which incident to 𝐴. In the other hand, the vertex π‘₯𝑖 and 𝑦𝑖 are adjacent, such that based on the edge labeling it can not receive the same weight. From this condition, such that we have π‘Žπ‘Ÿπ‘£π‘(ℱ𝑛) β‰₯ 3. Furthermore, to show the upper bound we construct the bijective function of edge labels. 𝑔(𝐴π‘₯𝑖 ) = 𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛 𝑔(π‘₯𝑖 𝑦𝑖 ) = 2𝑛 + 1 βˆ’ 𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛 𝑔(𝐴𝑦𝑖 ) = 2𝑛 + 𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛 From the edge labels above, we have the vertex weights in the following. 𝑀(π‘₯𝑖 ) = 2𝑛 + 1 𝑀(𝑦𝑖 ) = 4𝑛 + 1 𝑀(𝐴) = 3𝑛2 + 𝑛 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 ℱ𝑛is rainbow vertex antimagic coloring. Suppose that π‘₯, 𝑦 ∈ 𝑉(ℱ𝑛), refer to the vertex weight the rainbow vertex π‘₯ βˆ’ 𝑦 path is shown in Table 3. Table 3. The Rainbow Vertex of π‘₯ βˆ’ 𝑦 Path of ℱ𝑛 Case 𝒙 π’š Rainbow Vertex Coloring 𝒙 βˆ’ π’š 1 π‘₯𝑖 π‘₯𝑖 π‘₯𝑖 , 𝐴, π‘₯𝑖 2 π‘₯𝑖 𝑦𝑖 π‘₯𝑖 , 𝐴, 𝑦𝑖 3 𝑦𝑖 𝑦𝑖 𝑦𝑖 , 𝐴, 𝑦𝑖 4 𝑦𝑖 π‘₯𝑖 𝑦𝑖 , 𝐴, π‘₯𝑖 Hence, the vertex coloring of ℱ𝑛 is rainbow vertex antimagic coloring. Thus, we obtain π‘Ÿπ‘£π‘Žπ‘(ℱ𝑛) is 3 . ∎ Theorem 4 If 𝐹𝑛 be a fan graph 𝑛+1 and 𝑛 β‰₯ 3, then π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛 ) = 2 if 𝑛 ≑ 1(mod 2) and 2 ≀ π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛) ≀ 3 if 𝑛 ≑ 0(mod 2). Proof. Let 𝐹𝑛 be a fan graph with vertex set 𝑉(𝐹𝑛 ) = {𝐴, π‘₯1, π‘₯2, π‘₯3, β‹― , π‘₯𝑛} and edge set 𝐸(𝐹𝑛 ) = {𝐴π‘₯𝑖 : 1 ≀ 𝑖 ≀ 𝑛} βˆͺ {π‘₯𝑖 π‘₯{𝑖+1}: 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1}. The diameter of 𝐹𝑛 is 2. Based on Remark 1, we have π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛) β‰₯ π‘Ÿπ‘£π‘(𝐹𝑛 ) = π‘‘π‘–π‘Žπ‘š(𝐹𝑛) βˆ’ 1 = 2 βˆ’ 1 = 1. Since the vertex 𝐴 has degree of much greater than the others, it must have a different vertex weight than the others. The vertex weight of 𝐴 is the sum of labels of edges which incident to 𝐴. From this condition, such that we have π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛) β‰₯ 2. We divide into two cases to show the upper bound of the antimagic rainbow connection number of 𝐹𝑛 as follows. Case 1. For 𝐹𝑛 , 𝑛 ≑ 1(mod 2) To show the upper bound of π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛): 𝑛 ≑ 1(mod 2), we construct the bijective function of edge labels. 𝑔(π‘₯𝑖 π‘₯𝑖+1) = { 𝑖 2 , if 𝑖 ≑ 0(mod 2) 𝑛 + 𝑖 2 , if 𝑖 ≑ 1(mod 2) On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion Marsidi 70 𝑔(𝐴π‘₯𝑖 ) = { 2𝑛 βˆ’ 1, if 𝑖 = 𝑛 2𝑛 βˆ’ 𝑖 βˆ’ 1, if 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 From the edge labels above, we have the vertex weights in the following. 𝑀(π‘₯𝑖 ) = 5𝑛 βˆ’ 3 2 𝑀(𝐴) = 3𝑛2 βˆ’ 𝑛 2 From the vertex weights above, it is easy to see that the different weight is 2. Case 2. For 𝐹𝑛 , 𝑛 ≑ 0(mod 2) To show the upper bound of π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛): 𝑛 ≑ 0(mod 2), we construct the bijective function of edge labels. 𝑔(π‘₯𝑖 π‘₯𝑖+1) = { 𝑖 2 , if 𝑖 ≑ 0(mod 2) 𝑛 + 𝑖 βˆ’ 1 2 , if 𝑖 ≑ 1(mod 2) 𝑔(𝐴π‘₯𝑖 ) = { 2𝑛 βˆ’ 1, if 𝑖 = 𝑛 2𝑛 βˆ’ 𝑖 βˆ’ 1, if 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 From the edge labels above, we have the vertex weights in the following. 𝑀(π‘₯𝑖 ) = { 3𝑛 βˆ’ 2, if 𝑖 = 𝑛 5𝑛 2 βˆ’ 2, if 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 𝑀(𝐴) = 3𝑛2 βˆ’ 𝑛 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 𝐹𝑛 is rainbow vertex antimagic coloring. Suppose thatπ‘₯, 𝑦 ∈ 𝑉(𝐹𝑛 ), refer to the vertex weight the rainbow vertex π‘₯ βˆ’ 𝑦 path is shown in Table 4. Table 4. The Rainbow Vertex of π‘₯ βˆ’ 𝑦 Path of 𝐹𝑛 Case 𝒙 π’š Rainbow Vertex Coloring 𝒙 βˆ’ π’š 1 π‘₯𝑖 𝐴 π‘₯𝑖 , 𝐴 2 π‘₯𝑖 π‘₯𝑖 π‘₯𝑖 , 𝐴, π‘₯𝑖 Hence, the vertex coloring of 𝐹𝑛 is rainbow vertex antimagic coloring. Thus, we obtain π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛) = 2 if 𝑛 ≑ 1(mod 2) and 2 ≀ π‘Ÿπ‘£π‘Žπ‘(𝐹𝑛 ) ≀ 3 if 𝑛 ≑ 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 π‘Š17 satisfy the rainbow vertex antimagic coloring and rainbow vertex antimagic connection number of π‘Š17 is 2. On Rainbow Vertex Antimagic Coloring of Graphs: A New Notion Marsidi 71 Figure 2. The Illustration Rainbow Vertex Antimagic Coloring of π‘Š17 CONCLUSIONS We have obtained the exact values of rainbow vertex antimagic connection number of some connected graphs, namely path (𝑃𝑛 ), wheel (π‘Šπ‘›), friendship (ℱ𝑛), and fan (𝐹𝑛 ). However, since obtaining rainbow vertex antimagic connection number of graph is considered to be NP-complete problem, the characterization of the exact value of π‘Žπ‘Ÿπ‘£π‘(𝐺) 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. 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