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 rxk(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 rxk(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.
@article{bwmeta1.element.doi-10_7151_dmgt_1794,
author = {Xueliang Li and Ingo Schiermeyer and Kang Yang and Yan Zhao},
title = {Graphs with 4-Rainbow Index 3 and n - 1},
journal = {Discussiones Mathematicae Graph Theory},
volume = {35},
year = {2015},
pages = {387-398},
zbl = {1311.05032},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1794}
}
Xueliang Li; Ingo Schiermeyer; Kang Yang; Yan Zhao. Graphs with 4-Rainbow Index 3 and n − 1. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 387-398. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1794/
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