# Graphs with 4-Rainbow Index 3 and n − 1

Xueliang Li; Ingo Schiermeyer; Kang Yang; Yan Zhao

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 2, page 387-398
- ISSN: 2083-5892

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topXueliang Li, et al. "Graphs with 4-Rainbow Index 3 and n − 1." Discussiones Mathematicae Graph Theory 35.2 (2015): 387-398. <http://eudml.org/doc/271092>.

@article{XueliangLi2015,

abstract = {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.},

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 = {2},

pages = {387-398},

title = {Graphs with 4-Rainbow Index 3 and n − 1},

url = {http://eudml.org/doc/271092},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Xueliang Li

AU - Ingo Schiermeyer

AU - Kang Yang

AU - Yan Zhao

TI - Graphs with 4-Rainbow Index 3 and n − 1

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 2

SP - 387

EP - 398

AB - 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.

LA - eng

KW - rainbow S-tree; k-rainbow index; rainbow -tree; -rainbow index

UR - http://eudml.org/doc/271092

ER -

## References

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