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The strong 3-rainbow index of some certain graphs and its amalgamation

Awanis Zata Yumni, Salman A.N.M.

Opuscula Mathematica

2022

Vol. 42, no. 4

527--547

We introduce a strong k-rainbow index of graphs as modification of well-known k-rainbow index of graphs. A tree in an edge-colored connected graph G, where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let k be an integer with 2 ≤ k ≤ n. The strong k-rainbow index of G, denoted by srxk(G), is the minimum number of colors needed in an edge-coloring of G so that every k vertices of G is connected by a rainbow tree with minimum size. We focus on k = 3. We determine the strong 3-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong 3-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong 3-rainbow index of amalgamation of some graphs.