On the Study of Rainbow Antimagic Connection Number of Comb Product of Friendship Graph and Tree

Abstract

Given a graph G with vertex set V(G) and edge set E(G), for the bijective function f(V(G))→{1,2,⋯,|V(G)|}, the associated weight of an edge xy∈E(G) under f is w(xy)=f(x)+f(y). If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow x−y path if for every two edges xy,x′y′∈E(P) it satisfies w(xy)≠w(x′y′). The function f is called a rainbow antimagic labeling of G if there exists a rainbow x−y path for every two vertices x,y∈V(G). We say that graph G admits a rainbow antimagic coloring when we assign each edge xy with the color of the edge weight w(xy). 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). This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph Fn⊳Tm, where Fn is a friendship graph with order 2n+1 and Tm∈{Pm,Sm,Brm,p,Sm,m}, where Pm is the path graph of order m, Sm is the star graph of order m+1, Brm,p is the broom graph of order m+p and Sm,m is the double star graph of order 2m+2.