On the study of rainbow antimagic connection number of corona product of graphs

Abstract

Given that a graph G = (V, E). By an edge-antimagic vertex labeling of graph, we mean assigning labels on each vertex under the label function f : V → {1, 2, . . ., |V (G)|} such that the associated weight of an edge uv ∈ E(G), namely w(xy) = f(x) + f(y), has distinct weight. A path P in the vertex-labeled graph G is said to be a rainbow path if for every two edges xy, x′y′ ∈ E(P) satisfies w(xy) ≠ w(x′y′). The function f is called a rainbow antimagic labeling of G if for every two vertices x and y of G, there exists a rainbow x - y path. When we assign each edge xy with the color of the edge weight w(xy), thus we say the graph G admits a rainbow antimagic coloring. The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors induced from all edge weight of antimagic labeling. In this paper, we will study the rac(G) of the corona product of graphs. By the corona product of graphs G and H, denoted by G ⊙ H, we mean a graph obtained by taking a copy of graph G and n copies of graph H, namely H1, H2, ..., Hn, then connecting vertex vi from the copy of graph G to every vertex on graph Hi, i = 1, 2, 3, . . ., n. In this paper, we show the exact value of the rainbow antimagic connection number of Tn ⊙ Sm where Tn ∈ {Pn, Sn, Sn,p, Fn,3}.