Geodesic
Graphs with 4-Rainbow Index 3 and n − 1
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 387-398.

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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 called a rainbow tree if no two edges of T receive the same color. For a vertex set 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 every set S of k vertices of V (G) is called the k-rainbow index of G, denoted by r x_k (G). Notice that a lower bound and an upper bound of the k-rainbow index of a graph with order n is k − 1 and n − 1, respectively. Chartrand et al. got that the k-rainbow index of a tree with order n is n − 1 and the k-rainbow index of a unicyclic graph with order n is n − 1 or n − 2. Li and Sun raised the open problem of characterizing the graphs of order n with r x_k (G) = n − 1 for k ≥ 3. In early papers we characterized the graphs of order n with 3-rainbow index 2 and n − 1. In this paper, we focus on k = 4, and characterize the graphs of order n with 4-rainbow index 3 and n − 1, respectively.
Keywords: rainbow S-tree, k-rainbow index 
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Li, Xueliang; Schiermeyer, Ingo; Yang, Kang; Zhao, Yan. Graphs with 4-Rainbow Index 3 and n − 1. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 387-398. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a14/

[1] J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008).

[2] Q. Cai, X. Li and J. Song, Solutions to conjectures on the (k, ℓ)-rainbow index of complete graphs, Networks 62 (2013) 220-224. doi:10.1002/net.21513

[3] Y. Caro, A. Lev, Y. Roditty, Zs. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) R57.

[4] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98.

[5] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks 54 (2009) 75-81. doi:10.1002/net.20296

[6] G. Chartrand, S.F. Kappor, L. Lesniak and D.R. Lick, Generalized connectivity in graphs, Bull. Bombay Math. Colloq 2 (1984) 1-6.

[7] G. Chartrand, F. Okamoto and P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55 (2010) 360-367. doi:10.1002/net.20399

[8] L. Chen, X. Li, K. Yang and Y. Zhao, The 3-rainbow index of a graph, Discuss. Math. Graph Theory 35 (2015) 81-94. doi:10.7151/dmgt.1780

[9] P. Erdős and A. Gyárfás, A variant of the classical Ramsey problem, Combinatorica 17 (1997) 459-467. doi:10.1007/BF01195000

[10] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, New York, 2012).

[11] X. Li, Y. Shi and Y. Sun, Rainbow connections of graphs: A survey, Graphs Combin. 29 (2013) 1-38. doi:10.1007/s00373-012-1243-2

[12] X. Li, I. Schiermeyer, K. Yang and Y. Zhao, Graphs with 3-rainbow index n−1 and n − 2, Discuss. Math. Graph Theory 35 (2015) 105-120. doi:10.7151/dmgt.1783