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The strong 3-rainbow index of edge-amalgamation of some graphs

Abstract:

Let G be a nontrivial, connected, and edge-colored graph of order n ≥ 3, where adjacent edges may be colored the same. Let k be an integer with 2 ≤ k ≤ n. A tree T in G is a rainbow tree if no two edges of T are colored the same. For S ⊆ V (G), the Steiner distance d(S) of S is the minimum size of a tree in G containing S. An edge-coloring of G is called a strong k-rainbow coloring if for every set S of k vertices of G there exists a rainbow tree of size d(S) in G containing S. The minimum number of colors needed in a strong k-rainbow coloring of G is called the strong k-rainbow index srxk(G) of G. In this paper, we study the strong 3-rainbow index of edge-amalgamation of graphs. We provide a sharp upper bound for the srx3 of edge-amalgamation of graphs. We also determine the srx3 of edge-amalgamation of some graphs.