On rainbow antimagic coloring of some graphs

Abstract

Let G(V, E) be a connected and simple graphs with vertex set V and edge set E. Define a coloring c : E(G) → {1, 2, 3, …, k}, k ∈ N as the edges of G, where adjacent edges may be colored the same. If there are no two edges of path P are colored the same then a path P is a rainbow path. The graph G is rainbow connected if every two vertices in G has a rainbow path. A graph G is called antimagic if the vertex sum (i.e., sum of the labels assigned to edges incident to a vertex) has a different color. Since the vertex sum induce a coloring of their edges and there always exists a rainbow path between every pair of two vertices, we have a rainbow antimagic coloring. The rainbow antimagic connection number, denoted by rcA (G) is the smallest number of colors that are needed in order to make G rainbow connected under the assignment of vertex sum for every edge. We have found the exact value of the rainbow antimagic connection number of ladder graph, triangular ladder, and diamond.

On the rainbow antimagic coloring of vertex amalgamation of graphs

“…Theorem 1 [24] To proof lower bound of rac(H) can use rac(H) ≥ max{∆(H), rc(H)} Theorem 2 [12] If A is a almalgamation of fan graph, then rc(A) = 3.…”

“…First, we prove the correctness of the lower bound of rac(H 1 ) is rac(H 1 ) ≥ mn − m. We know that H 1 graph is classified as a tree graph. Based on Proposition 1 [6] and Theorem 1 [24], so that:…”

“…We know that H 3 graph is classified as a tree graph. Based on Proposition 1 [6] and Theorem 1 [24], so that:…”

“…A graph is said to be rainbow antimagic coloring 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 connection such that there is a rainbow path with all different edge weight [24]. The minimum number of colors for a rainbow path to exist with the conditional edge weights w(x) = w(y) for any two vertices x and y is the definition of the rainbow antimagic connection number and denoted by rac(G).…”

“…Intan K et al [16] found the rainbow antimagic connection number of releated wheel graph. Sulistiono et al [24] made a theorem to prove the lower bound of rac(G). More study on rainbow antimagic coloring can be seen at [1,4,9,22].…”