The 3-rainbow index of amalgamation of some graphs with diameter 2

Abstract

Let G = (V, E) be a nontrivial, connected, and edge-colored graph with n vertices, and let k be an integer with 2 ≤ k ≤ n. A tree T in G is a rainbow tree, if no two edges of T receive the same color. A k-rainbow coloring of G is an edge coloring of G having property that for every set S of k vertices of G, there exists a rainbow tree T such that S ⊆ V (T). The minimum number of colors needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by rx _k (G). The distance d(x,y) of two vertices x and y in G is the length of a shortest x — y path in G. The greatest distance between any two vertices in G is the diameter of G, denoted by diam(G). Let {G ₁, G ₂,…, G_t } be a finite collection of graphs and each graph G_i have a fixed vertex v _0i called a terminal. The amalgamation of G ₁, G ₂,…, G_t , denoted by Amal(G ₁, v _0l, G ₂, v ₀₂ …, G _t , v _0t ), is a graph obtained by taking all the \({G}_{i}^{^{\prime} }\)s and identifying their terminals. In case G_i ≅ G and v _0i = u, the amalgamation of G ₁, G ₂,…, G _t is denoted by Amal(G,t,u). In this paper, we determine the 3-rainbow index of amalgamation of some graphs Amal(G,t,u) with diam(G) = 2.