Local antimagic vertex total coloring on fan graph and graph resulting from comb product operation

Let G = (V,E) be a connected graph with |V|=n and |E|=m. A bijection f: V(G) U E(G) → {1,2,3, …,n + m} is called local antimagic vertex total coloring if for any two adjacent vertices u and v, wt(u)≠ wt(v), where wt(u) = ∑e∈E(u) f(e) + f(u), and E (u) is a set of edges incident to u. Thus any local antimagic vertex total labeling induces a proper vertex coloring of G where the vertex v is assigned the color wt (v). The local antimagic vertex total chromatic number χ1να t(G) is the minimum number of colors taken over all colorings induced by local antimagic vertex total. In this paper we investigate local antimagic vertex total coloring on fan graph (Fn) and graph resulting from comb product operation of Fn and F3 which denoted by Fnt> F3. We get two theorems related to the local antimagic vertex total chromatic number. First, χ1να t(Fn) = 3 where n> 3. Second, 3 < χ1ναt(Fn > F3) < 5 where n > 3.


Introduction
There are many topics in graph theory, one of them is coloring. In general, graph coloring is the giving of color to elements on the graph so that neighboring elements have a different color. Based on its element, graph coloring is divided into three kinds, namely vertex coloring, edge coloring and regional coloring. The detail of this theory can be read in [7] and [8].
The simple concept of graph coloring then has been developed into graph labeling. In general, graph labeling is the giving of labels, which is natural numbers, to the elements on the graph such as vertices or edges, or both [12]. Graph labeling is divided into two, namely magic labeling and antimagic labeling. Magic labeling on graph is labeling the elements on such that the sum of the labels of all elements incident with any vertex is the same [9]. On the contrary, if the sum of the labels of all elements incident with any two neighboring vertex is different its called antimagic labeling.
The research on antimagic labeling continues to be carried out by many researches. Dafik, et al. built the concept of super edge-antimagic total labeling [5,6]. Arumugam, et al. have the concept of local antimagic vertex coloring [4]. Agustin, et al. got the concept of local edge antimagic coloring [1] which then developed to super local edge antimagic total coloring [2,3,10,11]. Next, Putri, et al. built the concept of local vertex antimagic total coloring [13] which then continued by Kurniawati, et al. [12].
The concept of local vertex antimagic total coloring is explained as follows. Let = ( , ) be a connected graxph with | | = and | | = . A bijection : ( ) ∪ ( ) → { , , , . . . , + } is , and ( ) is a set of edges incident to . Thus any local antimagic vertex total labeling induces a proper vertex coloring of where the vertex is assigned the color ( ). The local antimagic vertex total chromatic number  ( ) is the minimum number of colors taken over all colorings induced by local vertex antimagic total labelings of [12].
In the previous research, we got the local antimagic vertex total coloring on some families tree [13] and graphs with homogeneous pendant vertex [12]. Based on this, the author conduct further research on local antimagic vertex total coloring on fan graph ( ) and graph resulting from comb product operation, namely ⊳ . Lemma 1.1 [13] If ( ) is a vertex coloring chromatic number of graph then ( ) ≥ ( )

Local Antimagic Vertex Total Coloring on Fan Graph
Before we got the local antimagic vertex total coloring of fan graph, we must know its vertex coloring chromatic number. Lemma 2.1 If is fan graph then the vertex coloring chromatic number of is ( ) = 3 Proof. Vertex coloring of graph is coloring the vertices in such that any two adjacent vertices in ( ) have diferent color. We have vertex set ( ) = { } ∪ { ; 1 ≤ ≤ } and edge set ( ) = { ; 1 ≤ ≤ } ∪ { +1 ; 1 ≤ ≤ − 1}. So, the color of vertex must be different than the color of vertex and the color of vertex must be different than the color of vertex +1 . If we gave a color for vertex , i.e: color 1, then the color of vertices , for = 1,2, … , , can not be color 1. Next if we gave vertex color 2 then the color of vertex +1 can not be color 2, for = 1,2, … , − 1. So, we have 3 colors for and this is the minimum color. Therefore ( ) = 3. ∎  It will be proven that function is local antimagic vertex total coloring by proving that is a bijection and neighboring vertices have different weights. First it will be proven that is a bijection. It is known that It is clear that range and codomain have the same cardinality so is a surjective function. Next, it will be proven that is an injection. For any , ∈ ( ) and ≠ applies ( ) ≠ ( ), so is an injective function. Since is surjection and injection, then is a bijection.
By function , we get the total vertex weight of fan graph ( ). The total vertex weight is obtained from the sum of vertex labels and edge labels. The total vertex weights are as follows: Based on the total vertex weight function, it will be shown that neighboring vertices have different weights. Vertex is adjacent to vertex and vertex is adjacent to vertex + , so it must have different weights. First, we assume that vertex has the same weight as vertex . It can be stated that This is a contradiction with the value of n ≥ 3 so the assumption is wrong. Thus vertex has a different weight from vertex . Next, we assume that vertex has the same weight as vertex + . If is odd then + is even and vice versa. So we have This is a contradiction with the value of n ≥ 3 so the assumption is wrong. Thus vertex has a different weight from vertex + . It can be concluded that neighboring vertices have different weights or ( ) ≠ ( ) and ( ) ≠ ( + ).
Since each neighboring vertex has different weights and is a bijection, it can be concluded that is local antimagic labeling. Then the vertices on fan graph ( ) are colored according to its total weight, which it is called local antimagic vertex total coloring. Because we have 3 total weights, then we also have 3 colors. Therefore, the local antimagic vertex total chromatic number of fan graph is ( ) ≤ . Next it will be shown that ( ) ≥ . Based on Lemma 1.1. and Lemma 2.1, we have ( ) ≥ ( ) = . Then ( ) ≥ . Because ( ) ≤ and ( ) ≥ , it is proven that ( ) = . As an illustration, it is presented by Figure 1 which is the local antimagic vertex total coloring of .  It will be proven that function is the local antimagic vertex total coloring by proving that is a bijection and neighboring vertices have different weights. First it will be proven that is a bijection. It is known that It is clear that range and codomain have the same cardinality so is a surjective function. Next, it will be proven that is an injection. For any , ∈ ( ) and ≠ applies ( ) ≠ ( ), so is an injective function. Since is surjection and injection, then is a bijection.
By function we get the total vertex weight of fan graph ( ). The total vertex weight is obtained from the sum of vertex labels and edge labels. The total vertex weights are as follows: Based on total vertex weight function above, it will be shown that neighboring vertices have different weights. Vertex is adjacent to vertex and vertex is adjacent to vertex + , so it must have different weights. First, we assume that vertex has the same weight as vertex . It can be stated that ( ) = ( ) or ( + ) = − = − . If the equation is solved, we get the value of = . This is a contradiction with the value of ≥ so the assumption is wrong. Thus vertex has a different weight from vertex . Next, we assume that vertex has the same weight as vertex + . If is odd then + is even and vice versa. So we have ( ) = ( + ) or − = − or = . This is a contradiction with the value of n ≥ 3 so the assumption is wrong. Thus vertex has a different weight from vertex + . It can be concluded that neighboring vertices have different weights or ( ) ≠ ( ) and ( ) ≠ ( + ).
Since each neighboring vertex has different weights and is a bijection, it can be concluded that is local antimagic labeling. Then the vertices on the fan graph ( ) are colored according to its total weight, which it is called local antimagic vertex total coloring. Because we have 3 total weights, then we also have 3 colors. Therefore, the local antimagic vertex total chromatic number of fan graph is ( ) ≤ . Next it will be shown that ( ) ≥ . Based on Lemma 1.1. and Lemma 2.1, we have ( ) ≥ ( ) = . Then ( ) ≥ . Because ( ) ≥ and ( ) ≤ , it is proven that ( ) = . As an illustration, it is presented by Figure 2 which is the local antimagic vertex total coloring of .

Local Antimagic Vertex Total Coloring on Fan Graph with Comb Product Operation
In this section, we got the local antimagic vertex total coloring of fan graph with comb product operation that is ⊳ 3 . Based on the definition of comb product operation, the obtained result of observation is as follows: Observation 3.1 Let ⊳ 3 be a graph resulting from comb product operation by taking one copy of and | ( )| copies of 3 and grafting the -th copy of 3 at the center vertex to -th vertex of . is 4( + 1) and edge set cardinality of ⊳ 3 is 7 + 4.
As an illustration, it is presented by Figure 3 which is the example of fan graph with comb product operation 5 ⊳ 3 . Before we got the local antimagic vertex total coloring of fan graph with comb product operation, we must know its vertex coloring chromatic number.   +1 ) and ( +1 ). If we gave a color for vertex , i.e: color 1, then the color of vertices , for = 1,2, … , , can not be color 1. Next if we gave vertex color 2 then the color of vertex +1 can not be color 2, for = 1,2, … , − 1, i.e: color 3. If vertex has color 1 then we can give color 2 and color 3 to vertice and +1 respectively. Next, if vertex has color 2 then we can give color 1 and color 3 to vertice and +1 respectively. And last, if vertex has color 3 then we can give color 1 and color 2 to vertice and +1 respectively. So, we have 3 colors for ⊳ 3 , and this is the minimum color. Therefore ( ⊳ 3 ) = 3. ∎ It will be proven that is the local antimagic vertex total coloring by proving that is a bijection and neighboring vertices have different weights. First it will be proven that is a bijection. It is known that = {1,2,3, . . . , 2 − 1} ∪ {2 , 2 + 1,2 + 2, . . . , 3 } ∪ {3 + 1,3 + 2, … ,5 + 2} ∪ {5 + 3, 5 + 4, … ,7 + 4} ∪ {7 + 5,7 + 6, … , 11 + 8} so the range of is = {1,2,3, . . . , 11 + 8}. Next, it will be proven that is an injection. For any , ∈ ( ) and ≠ applies ( ) ≠ ( ), so is an injective function. Since is surjection and injection, then is a bijection. By function we get the total vertex weight of graph . The total vertex weight is obtained from the sum of vertex labels and edge labels. The total vertex weights are as follows: = 26 + 17 if is even. If the equations is solved, we can not get the value of ≥ 3 so the assumption is wrong. Thus vertex has a different weight from vertex +1 and .
Next, vertex is adjacent to vertex +1 . We assume that ( ) = ( +1 ) or 19 + 11 = 26 + 17. If the equations is solved, we can not get the value of ≥ 3 so the assumption is wrong. Thus vertex has a different weight from vertex +1 .
Last, vertex is adjacent to vertex +1 . We assume that ( ) = ( +1 ) or 19 + 11 = 26 + 17. If the equations is solved, we can not get the value of ≥ 3 so the assumption is wrong. Thus vertex has a different weight from vertex +1 .
Since each neighboring vertex has different weights and is a bijection so it can be concluded that is local antimagic labeling. Then vertices on graph are colored based on their total vertex weights. This is called the local antimagic vertex total coloring. Because we have 5 values of total weights, then we also have 5 colors.. Therefore, the local antimagic vertex total chromatic number of graph is ( ) ≤ 5. Next it will be shown that ( ) ≥ 3. Based on Lemma 1.1. and Lemma 3.1, we have ( ) ≥ ( ) = 3. Then ( ) ≥ 3. Because ( ) ≤ 5 and ( ) ≥ 3, it is proven that 3 ≤ ( ) ≤ 5. As an illustration, it is presented by Figure 4 which is the local antimagic vertex total coloring of 5 ⊳ 3 .