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2015 | 35 | 2 | 387-398
Tytuł artykułu

Graphs with 4-Rainbow Index 3 and n − 1

Treść / Zawartość
Warianty tytułu
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
Wydawca
Rocznik
Tom
35
Numer
2
Strony
387-398
Opis fizyczny
Daty
wydano
2015-05-01
otrzymano
2014-01-14
poprawiono
2014-05-22
zaakceptowano
2014-06-16
online
2015-04-18
Twórcy
autor
  • Center for Combinatorics and LPMC-TJKLC Nankai University Tianjin 300071, China, lxl@nankai.edu.cn
autor
autor
Bibliografia
  • [1] J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008).
  • [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]
  • [3] Y. Caro, A. Lev, Y. Roditty, Zs. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) R57.
  • [4] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98.
  • [5] G. Chartrand, G.L. Johns, K.A. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks 54 (2009) 75-81. doi:10.1002/net.20296[WoS][Crossref]
  • [6] G. Chartrand, S.F. Kappor, L. Lesniak and D.R. Lick, Generalized connectivity in graphs, Bull. Bombay Math. Colloq 2 (1984) 1-6.
  • [7] G. Chartrand, F. Okamoto and P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55 (2010) 360-367. doi:10.1002/net.20399[Crossref][WoS]
  • [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]
  • [9] P. Erdős and A. Gyárfás, A variant of the classical Ramsey problem, Combinatorica 17 (1997) 459-467. doi:10.1007/BF01195000[Crossref]
  • [10] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, New York, 2012).
  • [11] 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[Crossref][WoS]
  • [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]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1794
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