Graphs with 4-Rainbow Index 3 and n − 1

• Autorzy: Xueliang Li , Ingo Schiermeyer , Kang Yang , Yan Zhao
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

rainbow S-tree; k-rainbow index

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2015
• Tom: 35
• Numer: 2
• Strony: 387-398

Daty

wydano

2015-05-01

otrzymano

2014-01-14

poprawiono

2014-05-22

zaakceptowano

2014-06-16

online

2015-04-18

Twórcy

autor

Xueliang Li

• Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China

autor

Ingo Schiermeyer

• Institut für Diskrete Mathematik und Algebra Technische Universität Bergakademie Frieiberg 09596 Freiberg, Germany

autor

Kang Yang

• Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China

autor

Yan Zhao

• Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China

Bibliografia

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• [2] Q. Cai, X. Li and J. Song, Solutions to conjectures on the (k, ℓ)-rainbow index of complete graphs, Networks 62 (2013) 220-224. doi:10.1002/net.21513[Crossref][WoS]
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• [4] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98.
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• [8] 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[WoS][Crossref]
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• [10] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, New York, 2012).
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• [12] X. Li, I. Schiermeyer, K. Yang and Y. Zhao, Graphs with 3-rainbow index n−1 and n − 2, Discuss. Math. Graph Theory 35 (2015) 105-120. doi:10.7151/dmgt.1783 [Crossref][WoS]

Identyfikatory

DOI

10.7151/dmgt.1794