# Graphs with 3-Rainbow Index n − 1 and n − 2

• Volume: 35, Issue: 1, page 105-120
• ISSN: 2083-5892

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## Abstract

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Let G = (V (G),E(G)) be a nontrivial connected graph of order n 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 set S ⊆ V (G), a tree connecting 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 rxk(G), where k is an integer such that 2 ≤ k ≤ n. Chartrand et al. got that the k-rainbow index of a tree is n−1 and the k-rainbow index of a unicyclic graph is n−1 or n−2. So there is an intriguing problem: Characterize graphs with the k-rainbow index n − 1 and n − 2. In this paper, we focus on k = 3, and characterize the graphs whose 3-rainbow index is n − 1 and n − 2, respectively.

## How to cite

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Xueliang Li, et al. "Graphs with 3-Rainbow Index n − 1 and n − 2." Discussiones Mathematicae Graph Theory 35.1 (2015): 105-120. <http://eudml.org/doc/271224>.

@article{XueliangLi2015,
abstract = {Let G = (V (G),E(G)) be a nontrivial connected graph of order n 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 set S ⊆ V (G), a tree connecting 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 rxk(G), where k is an integer such that 2 ≤ k ≤ n. Chartrand et al. got that the k-rainbow index of a tree is n−1 and the k-rainbow index of a unicyclic graph is n−1 or n−2. So there is an intriguing problem: Characterize graphs with the k-rainbow index n − 1 and n − 2. In this paper, we focus on k = 3, and characterize the graphs whose 3-rainbow index is n − 1 and n − 2, respectively.},
author = {Xueliang Li, Ingo Schiermeyer, Kang Yang, Yan Zhao},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {rainbow S-tree; k-rainbow index; rainbow -tree; -rainbow index},
language = {eng},
number = {1},
pages = {105-120},
title = {Graphs with 3-Rainbow Index n − 1 and n − 2},
url = {http://eudml.org/doc/271224},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Xueliang Li
AU - Ingo Schiermeyer
AU - Kang Yang
AU - Yan Zhao
TI - Graphs with 3-Rainbow Index n − 1 and n − 2
JO - Discussiones Mathematicae Graph Theory
PY - 2015
VL - 35
IS - 1
SP - 105
EP - 120
AB - Let G = (V (G),E(G)) be a nontrivial connected graph of order n 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 set S ⊆ V (G), a tree connecting 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 rxk(G), where k is an integer such that 2 ≤ k ≤ n. Chartrand et al. got that the k-rainbow index of a tree is n−1 and the k-rainbow index of a unicyclic graph is n−1 or n−2. So there is an intriguing problem: Characterize graphs with the k-rainbow index n − 1 and n − 2. In this paper, we focus on k = 3, and characterize the graphs whose 3-rainbow index is n − 1 and n − 2, respectively.
LA - eng
KW - rainbow S-tree; k-rainbow index; rainbow -tree; -rainbow index
UR - http://eudml.org/doc/271224
ER -

## References

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1. [1] J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008).
2. [2] G. Chartrand, G. Johns, K. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98. Zbl1199.05106
3. [3] G. Chartrand, F. Okamoto and P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55 (2010) 360-367. doi:10.1002/net.20339[WoS][Crossref] Zbl1205.05085
4. [4] 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[Crossref]
5. [5] Y. Caro, A. Lev, Y. Roditty, Zs. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) #R57. Zbl1181.05037
6. [6] G. Chartrand, S. Kappor, L. Lesniak and D. Lick, Generalized connectivity in graphs, Bull. Bombay Math. Colloq. 2 (1984) 1-6.
7. [7] G. Chartrand, G. Johns, K. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks 54 (2009) 75-81. doi:10.1002/net.20296[Crossref][WoS] Zbl1205.05124
8. [8] 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[WoS][Crossref] Zbl1258.05058
9. [9] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, 2012). Zbl1250.05066

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