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

Xueliang Li; Ingo Schiermeyer; Kang Yang; Yan Zhao

Discussiones Mathematicae Graph Theory (2015)

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

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topXueliang 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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