On the study of rainbow antimagic connection number of comb product of tree and complete bipartite graph

Rainbow antimagic coloring is the combination of antimagic labeling and rainbow coloring. The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by rac(G). Given a graph G with vertex set V (G) and edge set E(G), the function f from V (G) to {1, 2,⋯, |V (G)|} is a bijective function. The associated weight of an edge yz ∈ E(G) under f is w(yz) = f(y) + f(z). A path R in the vertex-labeled graph G is said to be a rainbow y - z path if for any two edges yz, y ′ z′ ∈ E(R) it satisfies w(yz) ≠ w (y′z′). The function f is called a rainbow antimagic labeling of G if there exists a rainbow y - z path for every two vertices y, z ∈ V (G). When we assign each edge yz with the color of the edge weight w(yz), we say the graph G admits a rainbow antimagic coloring. In this paper, we show the new lower bound and exact value of the rainbow antimagic connection number of comb product of any tree and complete bipartite graph.