The purpose of this study is to develop rainbow antimagic coloring. This study is a combination of two notions, namely antimagic and rainbow concept. If every vertex of graph G is labeled with the antimagic labels and then edge weight of antimagic labels are used to assign a rainbow coloring. The minimum number of colors for a rainbow path to exist with the condition satisfying the edge weights w(x) ≠ w(y) for any two vertices x and y is the definition of the rainbow antimagic connection number rac(G). In this study, we use connected graphs and simple graphs in obtaining the rainbow antimagic connection number. This paper will explain the rainbow antimagic coloring on some graphs and get their formula of the rainbow antimagic connection number. We have obtained rac(G) where G is vertex amalgamation of graphs, namely path, star, broom, paw, fan, and triangular book graph.
Let G be a graph which has V(G) as a vertex set and E(G) as an edge set. The G graph contain subgraph H that isomorphic with Hj, j = 1, 2, … s, if every e ∈ E(G) include only in one of the edge set of H subgraph (e ∈ E(H)). The total l-labeling is a graph labeling that give label positive integer number until l into vertices and edges. The total H-irregular labeling is a total l-labeling with condition that the sum of vertex labels and edge labels in two distinct subgraphs H 1 and H 2 isomorphic to H is different. We define H-weight as wtφ (H) = ∑ v∈V(H) φ(v) + ∑ e∈E(H) φ(e), for the subgraph H ⊆ G under the total l-labeling (φ). The total H-irregularity strength of G graph (ths(G, H)) is the smallest l value in label of G graph has total H-irregular labeling. We used plane graphs such as triangular ladder graph and windmill graph.
Let χ(G) be a chromatic number of proper coloring on G. For an injection f : V (G) → {0, 2, . . . ; 2kυg } and f : E(G) → }1, 2, . . . , ke }, where k = max{ke , 2kχ } for kυ , ke are natural number. The associated weight of a vertex u, υ ∈ V (G) under f is w(u) = f(u) + ∑ uυ∈E(G)f(uυ). The function f is called a local vertex irregular reflexive k-labeling if every two adjacent vertices has distinct weight. When we assign each vertex of G with a color of the vertex weight w(uυ), thus we say the graph G admits a local vertex irregular reflexive coloring. The smallest number of vertex weights needed to color the vertices of G such that no two adjacent vertices share the same color is called a local vertex irregular reflexive chromatic number, denoted by χirvs (G). Furthermore, the minimum k required such that χlrvs (G) = χ(G) is called a local reflexive vertex color strength, denoted by lrvcs(G). In this paper, we will obtain the lrvcs(G) and characterize the existence of a graph with given its local reflexive vertex color strength.
We define graph G as a nontrivial, finite, connected graph which contains vertex set V (G), edge set E(G), and face set F (G). We also define g as bijective function that mapping vertex, edge, and face labeling to natural number which starting from 1 until |V(G)| for vertex label, from |V(G)| + 1 until |V(G)| + |E(G)| for edge label, and the last for face label from |V (G)| + |E(G)| + 1 until |V (G)| + | E (G)| + |F(G)|. If there are different weights in any neighboring two faces f1 and f2 has w(f1) = w(f2) for f1, f2 G F (G), so g is considered a local super anti-magic total face labeling. A proper face coloring from local super anti-magic total face labeling caused by assigns the color of face weights to local super anti-magic total face coloring. The minimum number of colors needed for local super anti-magic total face coloring is called The chromatic number of the local super anti-magic total face coloring. γlatf (G) can be denoted as the chromatic number of the local super anti-magic total face coloring. Encryption keys can possibly be created from the result of local super anti-magic total face coloring that can be used to construct a modified Affine cipher and Cipher Feedback Mode. As a result, we have one the orem for the chromatic number of local super anti-magic total face coloring and two algorithms for establishing super anti-magic total face coloring on shackle graphs in Cipher Feedback Mode.
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