On rainbow antimagic coloring of some special graph

Abstract

Let G = (V, E) be a connected and simple graphs with vertex set V and edge set E. A coloring of graph G is rainbow connected if there is a rainbow path that connects each two vertices of graph G. The minimum k such that G has a rainbow-connected using k colors of the edges of G is the rainbow connection number rc(G) of G. A graph with a bijective mapping f : E → {1, 2, …, |E|}. The sums of each paired vertex has distinct value, defined as ∑ e ∈E(v)f(e). Thus, the function of G clearly an antimagic labeling if the sums of each paired vertex has distinct value. It is clear that rainbow antimagic connection number is the smallest number of colors which are needed to make G rainbow connected, denoted by rcA (G). A bijection function f : E → {1, 2, …, |E|} is called a rainbow antimagic labeling if there is a rainbow path between every pair of vertices and for each edge e = uv ∈ E(G), the weight w(e) = f(u) + f(v). A graph G is rainbow antimagic if G has a rainbow antimagic labeling. In this paper, we will analyze the rainbow antimagic coloring of related book graph.