Graphs with 3-Rainbow Index n − 1 and n − 2
Xueliang Li ; Ingo Schiermeyer ; Kang Yang ; Yan Zhao
Discussiones Mathematicae Graph Theory, Tome 35 (2015), p. 105-120 / Harvested from The Polish Digital Mathematics Library
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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.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:271224 
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     zbl = {1307.05051},
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Xueliang Li; Ingo Schiermeyer; Kang Yang; Yan Zhao. Graphs with 3-Rainbow Index n − 1 and n − 2. Discussiones Mathematicae Graph Theory, Tome 35 (2015) pp. 105-120. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1783/

[1] J.A. Bondy and U.S.R. Murty, Graph Theory (GTM 244, Springer, 2008).

[2] G. Chartrand, G. Johns, K. McKeon and P. Zhang, Rainbow connection in graphs, Math. Bohem. 133 (2008) 85-98. | Zbl 1199.05106

[3] G. Chartrand, F. Okamoto and P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55 (2010) 360-367. doi:10.1002/net.20339[WoS][Crossref] | Zbl 1205.05085

[4] 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[Crossref]

[5] Y. Caro, A. Lev, Y. Roditty, Zs. Tuza and R. Yuster, On rainbow connection, Electron. J. Combin. 15 (2008) #R57. | Zbl 1181.05037

[6] G. Chartrand, S. Kappor, L. Lesniak and D. Lick, Generalized connectivity in graphs, Bull. Bombay Math. Colloq. 2 (1984) 1-6.

[7] G. Chartrand, G. Johns, K. McKeon and P. Zhang, The rainbow connectivity of a graph, Networks 54 (2009) 75-81. doi:10.1002/net.20296[Crossref][WoS] | Zbl 1205.05124

[8] 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[WoS][Crossref] | Zbl 1258.05058

[9] X. Li and Y. Sun, Rainbow Connections of Graphs (Springer Briefs in Math., Springer, 2012). | Zbl 1250.05066