Abstract
Let G = (V, E) be a nontrivial, finite, and connected graph. A tree T in an edge-colored graph is said to be rainbow tree, if no two edges on the tree have the same color. An edge-coloring of G is called 3-rainbow coloring, if for any three vertices in G there exists a rainbow tree connecting them. The 3-rainbow index of G, denoted by rx3(G), is the minimum number of colors that are needed in a 3-rainbow coloring of G. The join of G1 and G2, denoted by G = G1 + G2, is the graph with vertex set V(G1) ∪ V(G2) and edge set E(G1) ∪ E(G2) ∪ {uv|u ∊ V(G1),v ∊ V(G2)}. In this paper, we determine rx3(G) where G is the join of a graph with a trivial graph.
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