# The 3-Rainbow Index of a Graph

Lily Chen; Xueliang Li; Kang Yang; Yan Zhao

Discussiones Mathematicae Graph Theory (2015)

- Volume: 35, Issue: 1, page 81-94
- ISSN: 2083-5892

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topLily Chen, et al. "The 3-Rainbow Index of a Graph." Discussiones Mathematicae Graph Theory 35.1 (2015): 81-94. <http://eudml.org/doc/271237>.

@article{LilyChen2015,

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 a rainbow tree if no two edges of T receive the same color. For a vertex subset 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 each k-subset S of V (G) is called the k-rainbow index of G, denoted by rxk(G). In this paper, we first determine the graphs of size m whose 3-rainbow index equals m, m − 1, m − 2 or 2. We also obtain the exact values of rx3(G) when G is a regular multipartite complete graph or a wheel. Finally, we give a sharp upper bound for rx3(G) when G is 2-connected and 2-edge connected. Graphs G for which rx3(G) attains this upper bound are determined.},

author = {Lily Chen, Xueliang Li, Kang Yang, Yan Zhao},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {rainbow tree; S-tree; k-rainbow index; -tree; -rainbow index},

language = {eng},

number = {1},

pages = {81-94},

title = {The 3-Rainbow Index of a Graph},

url = {http://eudml.org/doc/271237},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Lily Chen

AU - Xueliang Li

AU - Kang Yang

AU - Yan Zhao

TI - The 3-Rainbow Index of a Graph

JO - Discussiones Mathematicae Graph Theory

PY - 2015

VL - 35

IS - 1

SP - 81

EP - 94

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 a rainbow tree if no two edges of T receive the same color. For a vertex subset 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 each k-subset S of V (G) is called the k-rainbow index of G, denoted by rxk(G). In this paper, we first determine the graphs of size m whose 3-rainbow index equals m, m − 1, m − 2 or 2. We also obtain the exact values of rx3(G) when G is a regular multipartite complete graph or a wheel. Finally, we give a sharp upper bound for rx3(G) when G is 2-connected and 2-edge connected. Graphs G for which rx3(G) attains this upper bound are determined.

LA - eng

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

UR - http://eudml.org/doc/271237

ER -

## References

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