On the RACN of the comb product of the cycle C_3 with path P_n and broom Br_(n,m)

Abstract

The combination of rainbow coloring and anti-magic labeling is known as Rainbow Antimagic Coloring (RAC). The Rainbow Antimagic Connection Number (RACN) of a graph G is the smallest number of colors induced by all edge weights under an antimagic labeling, symbolized as rac(G) A graph is said to have rainbow antimagic connectivity if for every pir of vetices x∈V(G), there exits a rainbow antimagic path, wherin all edge weights along the path are distinct. Let G be a graph with vertex set V(G) and edge set E(G). A bijective function f from V(G) to {1,2,…,|V(G)|} is applied, wherein the weight of the edge uv∈E(G) is defined as w(uv) under f which is obtained from w(xv)=f(x)+f(v). A rainbow path x-v is a path in a vertex-labeled graph G if for any two edges xv,x' v'∈E(P) the path satisfies w(xv)≠w(x'v'). If there is a rainbow x-v path P for every two vertices x,v∈V(G) then the function f is called a rainbow antimagic labeling of G. A graph G we say has an RAC, if we assign each edge xv with an edge weight color w(xv). In this paper, we present the RACN of the comb product of cycle C_3 with path P_n and broom Br_(n,m) symbolized by C_3⊳ P_n and C_3⊳Br_(n,m). A comb operation on a graph G, symbolized as G⊳H, is a graph product wherein each vertex of G is replaced by a copy of H, maintaining the structure of G. This operation helps construct new classes of graphs with specific connectivity and labeling properties.