Geodesic
The 3-Rainbow Index of a Graph
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 81-94

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Let G be a nontrivial connected graph with an edge-coloring c : E(G) → 1, 2, . . ., q, q ∈ ℕ, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex subset S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a rainbow S-tree for each k-subset S of V(G) is called the k-rainbow index of G, denoted by rx_k(G). In this paper, we first determine the graphs of size m whose 3-rainbow index equals m, m − 1, m − 2 or 2. We also obtain the exact values of rx_3(G) when G is a regular multipartite complete graph or a wheel. Finally, we give a sharp upper bound for rx_3(G) when G is 2-connected and 2-edge connected. Graphs G for which rx_3(G) attains this upper bound are determined.
Keywords: rainbow tree, S-tree, k-rainbow index 
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Chen, Lily; Li, Xueliang; Yang, Kang; Zhao, Yan. The 3-Rainbow Index of a Graph. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 1, pp. 81-94. http://geodesic.mathdoc.fr/item/DMGT_2015_35_1_a6/