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

Abstract

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 [Formula: see text], denoted by [Formula: see text]. Given a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], the function [Formula: see text] from [Formula: see text] to [Formula: see text] is a bijective function. The associated weight of an edge [Formula: see text] under [Formula: see text] is [Formula: see text]. A path [Formula: see text] in the vertex-labeled graph [Formula: see text] is said to be a rainbow [Formula: see text] path if for any two edges [Formula: see text] it satisfies [Formula: see text]. The function [Formula: see text] is called a rainbow antimagic labeling of [Formula: see text] if there exists a rainbow [Formula: see text] path for every two vertices [Formula: see text]. When we assign each edge [Formula: see text] with the color of the edge weight [Formula: see text], we say the graph [Formula: see text] 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.